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IIT JEE Super Course in Mathematics ALGEBRA II

Trishna Knowledge Systems

Chapter 3

lermutations, Combinations and Binomial Theorem - all with Video Answers

Educators

Chapter Questions

01:28

Problem 1

How many numbers each of 4 -digits can be formed with the digits $0,1,3,5,7,9$ if no digit is to occur more than once in each number? Find their sum.

Shahab Ullah
Shahab Ullah
Numerade Educator
05:22

Problem 2

A candidate appears for an examination in which there are 4 papers with a maximum of $\mathrm{m}$ marks for each paper where $\mathrm{m}$ is a positive integer. Find the number of ways in which the candidate can score $2 \mathrm{~m}$ marks.

Nidhi Singhi
Nidhi Singhi
Numerade Educator
03:05

Problem 3

Show that $10^{\mathrm{n}}+3 \times 4^{\mathrm{n}+2}+5$ is divisible by 9 where, $\mathrm{n}$ is a positive integer.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:13

Problem 4

Prove that $\sum_{\mathrm{k}=0}^{\mathrm{n}} \frac{3^{\mathrm{k}+4} \times{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}}{\mathrm{k}+4}+\sum_{\mathrm{m}=0}^{3} \frac{{ }^{\mathrm{n}+4} \mathrm{C}_{\mathrm{m}}}{{ }^{\mathrm{n}+4} \mathrm{C}_{4}} \times 3^{\mathrm{m}}=\frac{4^{\mathrm{n}+4}}{\mathrm{n}+4} \mathrm{C}_{4}$.

Carson Merrill
Carson Merrill
Numerade Educator
04:06

Problem 5

If $\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots, \mathrm{C}_{\mathrm{n}}$ denote the coefficients of the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that
(i) $C_{0}^{2}+2 C_{1}^{2}+3 C_{2}^{2}+\ldots+(n+1) C_{n}^{2}=\frac{(n+2)(2 n-1) !}{n !(n-1) !}$
(ii) $\mathrm{C}_{0} \mathrm{C}_{\mathrm{r}}+\mathrm{C}_{1} \mathrm{C}_{\mathrm{r}+1}+\mathrm{C}_{2} \mathrm{C}_{\mathrm{r}+2}+\ldots .+\mathrm{C}_{\mathrm{n}-\mathrm{r}} \mathrm{C}_{\mathrm{n}}={ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}+\mathrm{r}}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
06:05

Problem 6

Find the number of non negative integral solutions of the equation $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\ldots+\mathrm{x}_{\mathrm{r}}=\mathrm{n}$, where, $\mathrm{n}$ is a positive integer $[\mathrm{OR}]$ Find the number of solutions of the equation $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\ldots+\mathrm{x}_{\mathrm{r}}=\mathrm{n}, \mathrm{n}$ being a positive integer and $0 \leq \mathrm{x}_{\mathrm{i}} \leq \mathrm{n}, \mathrm{i}=1,2, \ldots, \mathrm{n}$ and $\mathrm{x}_{\mathrm{i}}$ is an integer

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:14

Problem 7

Find the number of words that can be formed using the letters of the word AFFECTIONATE wherein the vowels and consonants have interchanged their places.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:37

Problem 8

A test paper in Mathematics has $\mathrm{n}$ questions. In that test $4^{\mathrm{k}}$ students gave correct answers to fewer than $\mathrm{k}$ questions, $k=1,2, \ldots n .$ If the total number of wrong answers is 87381 , find the value of $n$.

James Kiss
James Kiss
Numerade Educator
04:21

Problem 9

Given $\mathrm{u}_{\mathrm{n}}=7 \mathrm{u}_{\mathrm{n}-2}+6 \mathrm{u}_{\mathrm{n}-3}, \mathrm{n} \geq 3$ and $\mathrm{u}_{0}=9, \mathrm{u}_{1}=10, \mathrm{u}_{2}=32$, prove using mathematical induction that
$\mathrm{u}_{\mathrm{n}}=8(-1)^{\mathrm{n}}-3(-2)^{\mathrm{n}}+4(3)^{\mathrm{n}}, \mathrm{n}=0,1,2, \ldots \ldots$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
06:50

Problem 10

If $(1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{n} x_{n}$,
show that $\frac{n !}{x(x+1)(x+2) \ldots(x+n)}=\frac{C_{0}}{x}-\frac{C_{1}}{x+1}+\frac{C_{2}}{x+2} \ldots+(-1)^{r} \frac{C_{r}}{x+r}+\ldots+\frac{(-1)^{n} C_{n}}{x+n}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:25

Problem 11

Number of words that can be formed using the letters of the word "TRIUMPHANT", starting with T and ending with $\mathrm{T}$ is
(a) $10 !$
(b) $\frac{10 !}{2 !}$
(c) $8 !$
(d) ${ }^{10} \mathrm{P}_{8}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:54

Problem 12

Number of quadrilaterals that can be formed from a set of 12 points of which 4 of them are collinear is
(a) 70
(b) 462
(c) 224
(d) 392

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:47

Problem 13

${ }^{\mathrm{n}} \mathrm{C}_{5}+2{ }^{\mathrm{n}} \mathrm{C}_{4}+{ }^{\mathrm{n}} \mathrm{C}_{3}$ is
(a) $\left({ }^{n} C_{5}+{ }^{n} C_{3}\right)^{2}$
(b) ${ }^{(\mathrm{n}+1)} \mathrm{C}$
(c) ${ }^{(\mathrm{n}+1)} \mathrm{C}_{5}$
(d) ${ }^{n+2} C_{5}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:49

Problem 14

$\mathrm{n}$ books are arranged on a shelf so that two particular books are not next to each other. There were 480 arrangements altogether. Then the number of books on the shelf is
(a) 5
(b) 6
(c) 10
(d) 8

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:09

Problem 15

the total number of terms in the expansion of $\left(x^{2}+y^{2}\right)^{60}+\left(x^{2}-y^{2}\right)^{60}$, after simplification is
(a) 30
(b) 60
(c) 31
(d) 29

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:28

Problem 16

Statement 1
${ }^{15} \mathrm{C}_{4}+{ }^{15} \mathrm{C}_{5}+{ }^{16} \mathrm{C}_{6}={ }^{17} \mathrm{C}_{7}$
and
Statement 2
${ }^{n} C_{r-1}+{ }^{n} C_{r}={ }^{n+1} C_{r}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:19

Problem 17

Statement 1
Number of 3 -digits numbers that can be formed using the digits $0,1,4,7,9$ when repetitions are allowed is $100 .$ and
Statement 2
${ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}=\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2) \ldots . .(\mathrm{n}-\mathrm{r}+1)$

James Yang
James Yang
Numerade Educator
00:47

Problem 18

Statement 1
Number of ways of arranging 5 persons around a circular table so that 2 particular persons are always together is $12 .$ and
Statement 2 Number of ways of arranging $\mathrm{n}$ distinct objects around a circular table is $\frac{(\mathrm{n}-1) !}{2}$.

Clarissa Noh
Clarissa Noh
Numerade Educator
04:06

Problem 19

$\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \mathrm{C}_{3}, \ldots, \mathrm{C}_{\mathrm{n}}$ are the coefficients of powers of $\mathrm{x}$ in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$ where, $\mathrm{n}$ is a positive integer.
Statement 1 $\mathrm{C}_{1}+\mathrm{C}_{3}+\mathrm{C}_{5}+\ldots=2^{\mathrm{n}-1}$
and
Statement 2 $\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}+\ldots .+\mathrm{C}_{\mathrm{n}}=2^{\mathrm{n}}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:42

Problem 20

Statement 1 Term indepe ndent of $\mathrm{x}$ in the expansion of $\left(\mathrm{x}^{2}-\frac{3}{\mathrm{x}}\right)^{20}$ is the 11 th term.
and
Statement 2 If $n$ is even, the term independent of $x$ in the expansion of $\left(a x-\frac{b}{x}\right)^{n}$ is the $\left(\frac{n}{2}+1\right)$ th term.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:36

Problem 21

The total number of squares of all possible sizes that are available in a chess board is
(a) 204
(b) 64
(c) 56
(d) 48

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:48

Problem 22

The number of ways of choosing 4 squares each of $1 \mathrm{~cm}^{2}$ lying diagonally by on a Chess Board is
(a) 364
(b) 182
(c) 91
(d) 273

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:41

Problem 23

A rectangle is picked up from the board and it is seen to be of area $8 \mathrm{~cm}^{2}$. The number of ways in which this can happen is
(a) 8
(b) 172
(c) 43
(d) 86

Pammi Eswari
Pammi Eswari
Numerade Educator
01:59

Problem 24

A rectangle is picked up from the board and it is found that its perimeter is $10 \mathrm{~cm} .$ The number of ways in which this can happen is
(a) 41
(b) 82
(c) 164
(d) 328

Anas Venkitta
Anas Venkitta
Numerade Educator
01:18

Problem 25

3 squares are picked up from the board and it is found that their perimeters form an AP. The number of ways in which such triads of squares can be picked up is
(a) ${ }^{8} \mathrm{C}_{3}$
(b) 6
(c) 24
(d) 12

Manik Pulyani
Manik Pulyani
Numerade Educator
01:48

Problem 26

The number of ways of choosing at random 2 squares from the small squares of the chess board, having one corner in common
(a) 98
(b) 49
(c) 64
(d) 128

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
06:59

Problem 27

4 -digit numbers are formed using the digits $0,1,2,5,6,7,9$ with no repetition of digits in any of the numbers. Then
(a) Total number of numbers that can be formed is 720
(b) Number of numbers which are divisible by 5 is 240
(c) Number of numbers which are divisible by 4 is 168
(d) Number of numbers which are greater than 1500 is 680

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:23

Problem 28

12 men from a group of 32 soldiers are selected to form a regiment for general duty
(a) The number of such groups in which 2 particular soldiers $\mathrm{A}$ and $\mathrm{B}$ are there in the regiment formed is ${ }^{30} \mathrm{C}_{10}$
(b) The number of such groups in which 3 particular soldiers $\mathrm{C}, \mathrm{D}, \mathrm{E}$ are there in the regiment formed is ${ }^{29} \mathrm{C}_{9}$
(c) Number of such groups in which two particular soldiers $\mathrm{A}$ and $\mathrm{B}$ are not together, is ${ }^{32} \mathrm{C}_{10}$
(d) The number of ways of forming 4 regiments each containing 8 soldiers is $\frac{32 !}{(8 !)^{4}}$

Elizabeth Xu
Elizabeth Xu
Numerade Educator
05:30

Problem 29

Let $\left(2-x+3 x^{2}\right)^{6}=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots \ldots+a_{12} x^{12}$. Then,
(a) $a_{4}=1740$
(b) $a_{4}=3660$
(c) $\mathrm{a}_{0}+\mathrm{a}_{2}+\mathrm{a}_{4}+\ldots \ldots+\mathrm{a}_{12}=2^{5}\left(2^{6}+3^{6}\right)$
(d) $\mathrm{a}_{0}+\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}+\ldots \ldots+\mathrm{a}_{12}=2^{12}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
11:46

Problem 30

Column I
(a) ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{(\mathrm{n}+1)} \mathrm{C}_{\mathrm{r}+1}$
(b) $(n-r+1)\left[{ }^{n} C_{r-1}\right]$
(c) $(n+1)\left[{ }^{n} C_{r}\right]$
(d) $\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}\left[{ }^{(\mathrm{n}-1)} \mathrm{C}_{(\mathrm{r}-1)}\right]$
Column II
(p) $\mathrm{r} .{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$
(q) ${ }^{(\mathrm{r}+1)} \mathrm{C}_{2}\left[{ }^{(\mathrm{n}+1)} \mathrm{C}_{(\mathrm{r}+1)}\right]$
(r) $(n-r+1)\left[{ }^{(n+1)} C_{r}\right]$
(s) ${ }^{(\mathrm{n}+2)} \mathrm{C}_{(\mathrm{r}+1)}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:55

Problem 31

If ${ }^{\mathrm{n}+1} \mathrm{C}_{4}:^{\mathrm{n}-1} \mathrm{P}_{2}:: 25: 1$, then $\mathrm{n}$ is
(a) 25
(b) 24
(c) 15
(d) 26

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:07

Problem 32

The number of 5 digit binary numbers that can be formed, is
(a) $9 \times{ }^{9} \mathrm{P}_{5}$
(b) 16
(c) 25
(d) 36

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:49

Problem 33

If some or all of $\mathrm{n}$ things be taken at a time, the total number of combinations is
(a) $2^{\mathrm{n}}$
(b) $2^{n}-1$
(c) $\mathrm{n}(\mathrm{n}-1$
(d) ${ }^{2 \mathrm{n}} \mathrm{C}_{n}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:26

Problem 34

A hall contains 25 lamps, controlled by independent switches. Possible ways of illuminating the hall is
(a) $25 !$
(b) $2^{25}$
(c) $2^{25}-1$
(d) $25^{2}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:01

Problem 35

8 points are marked on the circumference of a circle at equal distances. How many squares can be drawn by joining
them?
$\begin{array}{lllll}\text { (a) }{ }^{8} \mathrm{P}_{4} & \text { (b) }{ }^{8} \mathrm{C}_{4} & \text { (c) } \frac{{ }^{8} \mathrm{C}_{4}}{2} & \text { (d) } 2\end{array}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:24

Problem 36

Number of possible words that can be formed by using the letters of the word "MALAYALAM" is
(a) 3780
(b) $\frac{9 !}{4 !}$
(c) $\frac{9 !}{2 ! \times 2 !}$
(d) $9 !$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:27

Problem 37

If the 4 th term in the expansion of $\left(\mathrm{kx}+\frac{2}{\mathrm{x}}\right)^{\mathrm{n}}$ is 20, the values of $\mathrm{k}$ and $\mathrm{n}$ are respectively
(a) $\frac{1}{2}$ and 6
(b) $\frac{1}{6}$ and 2.
(c) $\frac{1}{3}$ and
(d) $\frac{1}{2}$ and

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:36

Problem 38

The coefficient of $a^{2}$ in the expansion of $\left(1+a+a^{2}+a^{3}\right)^{10}$ is
(a) 50
(b) 110
(c) 55
(d) 550

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:56

Problem 39

The largest coefficient in the expansion of $(1+\mathrm{x})^{30}$ is
(a) ${ }^{30} \mathrm{C}_{6}$
(b) ${ }^{30} \mathrm{C}_{15}$
(c) ${ }^{30} \mathrm{C}_{29}$
(d) ${ }^{30} \mathrm{C}_{10}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:07

Problem 40

The sum of the coefficients in the expansion of $(7 x+3 y)^{4}$ is
(a) 10000
(b) 1000
(c) 2100
(d) 621

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:29

Problem 41

The value of $\mathrm{C}_{0}+\mathrm{C}_{2}+\mathrm{C}_{4}+\ldots$ in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$ is
(a) $2^{\mathrm{n}}$
(b) $2^{n+1}$
(c) $2^{n-1}$
(d) $2^{n}-1$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:25

Problem 42

If $\mathrm{C}_{\mathrm{r}}$ represents ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{C}_{0}^{2}+\mathrm{C}_{1}^{2}+\mathrm{C}_{2}^{2}+\ldots+\mathrm{C}_{\mathrm{n}}^{2}$ equals
(a) $\frac{(2 n) !}{(n !)^{2}}$
(b) $\frac{6}{7}$
(c) $\frac{1}{25}$
(d) $2 \mathrm{n} !$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:52

Problem 43

For $a, b \in R^{+}$, the expansion $(a-b x)^{-3}$ is valid only when
(a) $\frac{1}{6}$
(b) $|\mathrm{x}| \leq \frac{\mathrm{b}}{\mathrm{a}}$
(c) $|x|<b$
(d) $x>\frac{a}{b}$

Ahmad Reda
Ahmad Reda
Numerade Educator
01:44

Problem 44

In a conference, every delegate shakes hands with every other delegate. If 300 handshakes have been counted, then the number of delegates participated in the conference is
(a) 10
(b) 20
(c) 30
(d) 25

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:27

Problem 45

3 different books on Physics, 5 different books on Mathematics and 2 different books on Chemistry are to be arranged on a shelf. Number of different possible arrangements so that books on the same subject are kept together is
(a) $\frac{10 !}{5 ! \times 2 ! \times 3 !}$ $\begin{array}{lll}\text { (b) } 10 ! & \text { (c) } 3 ! \times 3 ! \times 5 ! \times 2 ! & \text { (d) } \frac{10 !}{2 ! \times 3 !}\end{array}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:02

Problem 46

A shoe-rack has five pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair is
(a) 80
(b) 160
(c) 200
(d) 240

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:18

Problem 47

In a class tournament where each participant was to play exactly one game with another, two players fell ill after playing 3 games each. If the total number of games played in the tournament was 84 , the number of participants was
(a) 12
(b) 15
(c) 24
(d) 30

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:22

Problem 48

$\frac{1}{3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots . .+\frac{1}{(2 n-1)(2 n+1)}=$
(a) $\frac{2 n}{2 n+1}$
(b) $\frac{n}{2 n+1}$
(c) $\frac{2 n}{2 n-1}$
(d) $\mathrm{C}_{0}{ }^{2}+2 \mathrm{C}_{1}^{2}+3 \mathrm{C}_{2}^{2}+\ldots .+(\mathrm{n}+1) \mathrm{C}_{\mathrm{n}}^{2}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:57

Problem 49

In the expansion of $(\sqrt{3}+1)^{10}$, the ratio of the 7 th term from the beginning to the 7 th term from the end is $1: 6$. Then,
$\mathrm{n}=$
(a) 6
(b) 7
(c) 9
(d) 12

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:14

Problem 50

If $10^{5}$ is the third term in the expansion of $\left(x+x^{\log _{10} x}\right)^{5}$, then $3 \log _{10} x+2\left(\log _{10} x\right)^{2}=$
(a) 6
(b) 5
(c) 4
(d) 10

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:20

Problem 51

If the coefficients of the 5 th, 6 th and 7 th terms of the expansion of $(1+\mathrm{x})^{\mathrm{n}}$ are in A.P, then the value of $\mathrm{n}$ may be
(a) 5
(b) 6
(c) 7
(d) 8

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:44

Problem 52

The largest term in the expansion of $(3+2 \mathrm{x})^{50}$ where $\mathrm{x}=\frac{1}{5}$ is
(a) 8 th term
(b) 7 th term
(c) 6 th term
(d) (b) or (c)

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:08

Problem 53

The sum of $n$ terms of the series $\frac{1}{n !}+\frac{1}{(n-1) !}+\frac{1}{2 !(n-2) !}+\frac{1}{3 !(n-3) !}+\ldots$ is
(a) $\frac{2^{n-1}}{(n-2) !}$
(b) $\frac{2^{n+1}}{n !}$
(c) $\frac{2^{n}}{n !}$
(d) $\frac{2^{n}}{n+1}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:25

Problem 54

The larger of $61^{30}$ and $59^{30}+60^{30}$ is
(a) cannot be determined
(b) both are equal
(c) $61^{30}$
(d) $59^{30}+60^{30}$

Aman Gupta
Aman Gupta
Numerade Educator
00:58

Problem 55

If $(1+x)^{14}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{14} x^{14}$, then $\frac{C_{1}}{C_{0}}+2 \times \frac{C_{2}}{C_{1}}+3 \times \frac{C_{3}}{C_{2}}+\ldots .+14 \times \frac{C_{14}}{C_{13}}$ equals
(a) 110
(b) 105
(c) 85
(d) 80

Gaurav Kalra
Gaurav Kalra
Numerade Educator
00:26

Problem 56

If $x$ is very large, $\sqrt{x^{2}+16}-\sqrt{x^{2}+9}=$
(a) $\frac{7}{2 \mathrm{x}}+\ldots$
(b) $\frac{7}{2} \mathrm{x}+\ldots$
(c) $\frac{7}{2 x}+x+\ldots$
(d) $-\frac{7}{2 \mathrm{x}}+\ldots \ldots$

Vivek Kumar
Vivek Kumar
Numerade Educator
02:38

Problem 57

The sum to infinity of the series $1+\frac{1}{3}+\frac{2 \cdot 5}{1 \cdot 2}\left(\frac{1}{6}\right)^{2}+\frac{2 \cdot 5 \cdot 8}{1 \cdot 2 \cdot 3}\left(\frac{1}{6}\right)^{3}+\ldots$
(a) $4^{1 / 3}$
(b) $2^{1 / 3}$
(c) $2 \sqrt{2}$
(d) $\sqrt{3}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:16

Problem 58

The coefficient of $x^{3}$ in the expansion of $1+(1+x)+(1+x)^{2}+(1+x)^{3}+\ldots+(1+x)^{n}$ where, $n$ is a positive integer greater than 4, is
(a) $\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3}$
(b) ${ }^{n} C_{2}+n$
(c) ${ }^{\mathrm{n}+1} \mathrm{C}_{4}$
(d) $\frac{C_{10}}{11}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
00:57

Problem 59

There are 12 questions in a question paper from which students have to answer only 8 ; the first question and the last question are compulsory. The number of ways a student can choose the right questions, is
(a) 210
(b) 330
(c) ${ }^{12} \mathrm{P}_{8} \times 8 !$
(d) 420

Hoan Nguyen
Hoan Nguyen
Numerade Educator
03:25

Problem 60

The number of diagonals of a polygon of 25 sides is
(a) 210
(b) 275
(c) 300
(d) 250

Christine Anacker
Christine Anacker
Numerade Educator
02:01

Problem 61

A child has 10 toys ( 5 identical and the remaining 5 all different). The number of ways in which the child can select 6 toys, is
(a) 31
(b) 32
(c) 33
(d) 64

Mir Afzal
Mir Afzal
Numerade Educator
03:48

Problem 62

A set of $n$ parallel equispaced straight lines in a plane intersect another set of n parallel equispaced lines which are in the same plane as the first set. Assume that interspacing of lines belonging to the second set is the same as that of the first set. Then the number of rhombuses so formed, is
(a) $\frac{2^{11}}{11}$
(b) $\frac{1}{10}$
(c) $\frac{n(n-1)(2 n-1)}{6}$
(d) $\frac{n(n+1)(2 n-1)}{6}$

NW
Nida Wasiq
Numerade Educator
00:53

Problem 63

The number of proper factors of the number $\mathrm{a}_{1}{ }^{3} \mathrm{a}_{2}{ }^{8} \mathrm{a}_{3}{ }^{15} \ldots \mathrm{a}_{\mathrm{n}}{ }^{(\mathrm{n}+1)^{2}-1}$ where, $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots \mathrm{a}_{\mathrm{n}}$ are distinct prime numbers, is
(a) $[(n+1) !]^{2}$
(b) $\frac{n(n+1)}{2}$
(c) $[(n-1) !]^{2}$
(d) $\frac{\mathrm{n}}{6}\left(2 \mathrm{n}^{2}+3 \mathrm{n}+7\right)$

Victor Salazar
Victor Salazar
Numerade Educator
03:33

Problem 64

If the words formed by the letters of the word "ANGER” are arranged in lexicographic order (dictionary order), then the rank of the word "RANGE" is
(a) 99
(b) 100
(c) 102
(d) 101

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
00:35

Problem 65

$999^{444}-4^{222}$ is divisible by
(a) 5
(b) 3
(c) 15
(d) 9

Brandon Fox
Brandon Fox
Numerade Educator
06:50

Problem 66

If $^{n-r} C_{x}+{ }^{(n-r)} C_{(x-1)}+{ }^{(n-r+1)} C_{(x-1)}+\ldots+{ }^{(n-2)} C_{(x-1)}+{ }^{(n-1)} C_{(x-1)} \geq n .{ }^{n} C_{(x+1)}$, then $x$ equals
(a) $n+1$
(b) $\mathrm{n}-1$
(c) $\mathrm{n}^{2}-1$
(d) $\mathrm{n}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:40

Problem 67

The letters of the word "SISTERS" are permuted. The number of words in which the 3 S's occupy the odd places, is
(a) 24
(b) 96
(c) 576
(d) 48

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
02:23

Problem 68

If $n \in N$, the number of solutions of the equation $1 !+2 !+3 !+\ldots+k !=n^{4}$, is
(a) 0
(b) 1
(c) 2
(d) 3

Lily An
Lily An
Numerade Educator
06:12

Problem 69

Coefficient of $\mathrm{x}^{98}$ in the expansion of $\sum_{\mathrm{k}=0}^{200}{ }^{200} \mathrm{C}_{\mathrm{k}}(\mathrm{x}-3)^{200-\mathrm{k}}(\mathrm{x}+2)^{\mathrm{k}}$ is
(a) $2^{98}$
(b) $-\left({ }^{200} \mathrm{C}_{98} \times 2^{98}\right)$
(c) ${ }^{200} \mathrm{C}_{98} \times 2^{98}$
(d) ${ }^{200} \mathrm{C}_{102} \times 2^{102}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:06

Problem 70

If $\mathrm{p}=\left(1+{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}+{ }^{\mathrm{n}} \mathrm{C}_{3}+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}\right)^{2}$ in the expansion of $(\mathrm{p}+\mathrm{q})^{\mathrm{n}}$ is equal to $\mathrm{q}=1+{ }^{2 \mathrm{n}} \mathrm{C}_{1}+{ }^{2 \mathrm{n}} \mathrm{C}_{2}+{ }^{2 \mathrm{n}} \mathrm{C}_{3}+\ldots+{ }^{2 \mathrm{n}} \mathrm{C}_{2 \mathrm{n}}$
in the expansion of $(p+q)^{n+3}$, then $n$ is
(a) 7
(b) 8
(c) 9
(d) 6

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:37

Problem 71

The number of positive integral solutions of $2 \mathrm{x}+3 \mathrm{y}+5 \mathrm{z}=20$, is
(a) 1
(b) 2
(c) 3
(d) 4

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:33

Problem 72

The number of ways of permuting the letters of the word "INFINITY" so that the two N's are never together, is
(a) 2519
(b) 2520
(c) 2521
(d) 3360

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:05

Problem 73

A total of 25 questions have to be set for an examination in Mathematics. The number of ways in which a teacher can allot the total marks of 100 to these questions given that each question is worth at least two mark, is
(a) ${ }^{74} \mathrm{C}_{50}$
(b) ${ }^{74} \mathrm{C}_{49}$
(c) ${ }^{75} \mathrm{C}_{50}$
(d) ${ }^{75} \mathrm{C}_{49}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:54

Problem 74

There are 15 points in a plane, no three of which are collinear exceptive for 5 points which line on are lines, 3 others on a second line and the remaining 6 on a third line, the lines being not concurrent. The total number of quadrilaterals that can be formed with these points as vertices, is
(a) 495
(b) 1001
(c) 225
(d) 720

Diwakar Mandilwar
Diwakar Mandilwar
Numerade Educator
02:29

Problem 75

A circular shelf holds 12 different books. The number of ways of choosing 4 books if no two adjacent books are chosen, is
(a) $7 ! \times{ }^{8} \mathrm{P}_{4}$
(b) $7 ! \times{ }^{8} \mathrm{C}_{4}$
(c) $\sqrt{3}\left\{(\sqrt{3}+1)^{20}-(\sqrt{3}-1)^{20}\right\}$
(d) $7 ! \times{ }^{7} \mathrm{P}_{4}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:12

Problem 76

The number of different strings of length 5 or more that could be formed using the letters of the word "BANANA", is
(a) 60
(b) 120
(c) 90
(d) 110

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:19

Problem 77

The last four digits of $7^{4000}$ are
(a) 1101
(b) 1001
(c) 1011
(d) 0001

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:20

Problem 78

One value of $\mathrm{x}$, for which the 7 th term in the expansion of $\frac{\mathrm{k}}{\ell}$ equals 168 , is
(a) 7
(b) 5
(c) 2
(d) 6

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:36

Problem 79

The number of non-negative integral solutions of $10 \leq \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4}+\mathrm{x}_{5} \leq 20$, is
(a) 9191
(b) 9911
(c) 9119
(d) 9991

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
01:16

Problem 80

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{5})^{256}$ is
(a) 32
(b) 33
(c) 34
(d) 35

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 81

The number of different signals that can be given by hoisting any number of flags from among 6 flags of different colours arranged in a row is
(a) 2040
(b) 1956
(c) 940
(d) 1965

Cheyenne Whinham
Cheyenne Whinham
Numerade Educator
02:10

Problem 82

The coefficient of $x^{3}$ in the expansion of $\left(x^{3}-\frac{1}{x^{4}}\right)^{15}$ is
(a) $-455$
(b) 455
(c) 5005
(d) $-5005$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:54

Problem 83

The number of terms in the expansion of $\left[(2 p+3 q)^{5} \cdot(2 p-3 q)^{5}\right]^{4}$ is
(a) 5
(b) 9
(c) 20
(d) 21

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
01:13

Problem 84

The fractional part of $\frac{7^{25}}{6}$ is
(a) 0
(b) $\frac{6}{7}$
(c) $\frac{1}{25}$
(d) $\frac{1}{6}$

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 85

If the unit digit of $27^{\mathrm{n}}-17^{\mathrm{n}}+8^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}$ is 2 , then the remainder when $\mathrm{n}$ is divided by 4 is
(a) 3
(b) 0
(c) 1
(d) 2

Grant Castaneda
Grant Castaneda
Numerade Educator
01:18

Problem 86

An approximate value of $(1.01)^{5}$ is
(a) $1.0001$
(b) $1.051$
(c) $1.51$
(d) $1.0000001$

Malika Singh
Malika Singh
Numerade Educator
01:23

Problem 87

A flower basket contains 8 flowers of which two are identical. Number of ways 3 flowers can be selected from the basket is
(a) 56
(b) 28
(c) 41
(d) 39

Manik Pulyani
Manik Pulyani
Numerade Educator
03:55

Problem 88

The number of ways in which 6 different prizes can be distributed among 3 children each receiving at least one prize is
(a) 270
(b) 720
(c) 540
(d) 620

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
01:07

Problem 89

The number of terms in the expansion of $(a+2 b+3 c)^{n}$ is 55 . Then $n$ is equal to
(a) 6
(b) 7
(c) 8
(d) 9

Aman Gupta
Aman Gupta
Numerade Educator
01:54

Problem 90

$\frac{1}{49^{n}}-{ }^{2 \mathrm{n}} \mathrm{C}_{1} \frac{8}{49^{\mathrm{n}}}+{ }^{2 \mathrm{n}} \mathrm{C}_{2} \frac{8^{2}}{49^{\mathrm{n}}}-\ldots \frac{8^{2 \mathrm{n}}}{49^{\mathrm{n}}}$ is equal to
(a) $\frac{1}{49^{n}}$
(b) $7^{\mathrm{n}}$
(c) 1
(d) 0

Nick Johnson
Nick Johnson
Numerade Educator
02:29

Problem 91

If $p+q=1$, then $\sum_{r=0}^{n} r \cdot{ }^{n} C_{r} p^{r} q^{n-r}$ is equal to
(a) 1
(b) $\mathrm{np}$
(c) npq
(d) 0

Wendi Zhao
Wendi Zhao
Numerade Educator
07:46

Problem 92

The constant term in the expansion of $(1+x)^{n}\left(1+\frac{1}{x}\right)^{n}$ is
(a) $\mathrm{C}_{0}^{2}+\mathrm{C}_{1}^{2}+\mathrm{C}_{2}^{2}+\ldots .+\mathrm{C}_{\mathrm{n}}^{2}$
(b) $\left(\mathrm{C}_{0}+\mathrm{C}_{2}+\mathrm{C}_{4}+\ldots .+\ldots\right)^{2}$
(c) $\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}+\ldots . \mathrm{C}_{\mathrm{n}}$
(d) $C_{0}{ }^{2}+2 C_{1}^{2}+3 C_{2}^{2}+\ldots .+(n+1) C_{n}^{2}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:36

Problem 93

Let $\mathrm{P}$ be the sum of the odd terms and $\mathrm{Q}$, the sum of the even terms in the expansion of $(\sqrt{3}+1)^{10}$. Then, $\mathrm{P}^{2}-\mathrm{Q}^{2}$ is
(a) 1
(b) $2^{5}$
(c) $2^{10}$
(d) $3^{10}$

Aman Gupta
Aman Gupta
Numerade Educator
02:57

Problem 94

The minimum number of people needed to guarantee that at least 7 of them have the same astrological sign, is
(a) 84
(b) 72
(c) 73
(d) 13

Amany Waheeb
Amany Waheeb
Numerade Educator
03:43

Problem 95

Two straight lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ pass through $\mathrm{A}$. Excluding $\mathrm{A}$ there are 4 points on $\mathrm{L}_{1}$ and 5 on $\mathrm{L}_{2} .$ The number of triangles that can be formed with these points as vertices, is
(a) 90
(b) 135
(c) 20
(d) 70

Bharat Bairagi
Bharat Bairagi
Numerade Educator
01:08

Problem 96

If $n \in N$ is a prime number, then the number of negative terms in the sequence $\left(x_{n}\right)$, where $x_{n}={ }^{(n+7)} C_{6}-\frac{7}{72}{ }^{(n+7)} P_{4}$, is
(a) 0
(b) 1
(c) 3
(d) infinite

Raushan Kumar
Raushan Kumar
Numerade Educator
02:31

Problem 97

The exponents of 3,2 and 5 in $300 !$ are in the ratio
(a) $4: 5: 2$
(b) $2: 4: 1$
(c) $4: 2: 1$
(d) $5: 6: 4$

Allison Knapp
Allison Knapp
Numerade Educator
03:19

Problem 98

The number of factors of $\mathrm{k}=\min [\mathrm{n} !(13-\mathrm{n}) !]$ is
(a) 540
(b) 135
(c) 270
(d) 90

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:31

Problem 99

Let $\mathrm{k}$ and $\mathrm{m}$ denote respectively the exponent of 3 in $207 !$ and the number of divisors of 315000 then
(a) $\mathrm{k}>\mathrm{m}$
(b) $\mathrm{k}<\mathrm{m}$
(c) $\mathrm{k}=\mathrm{m}$
(d) $\mathrm{k}=2 \mathrm{~m}$

Nick Johnson
Nick Johnson
Numerade Educator
02:16

Problem 100

If the 8 th term in the expansion of $\left[\mathrm{y}^{\log _{3} \mathrm{y}}+\mathrm{y}\right]^{9}$ is 2916, then y can be
(a) 81
(b) 3
(c) 9
(d) $\frac{1}{81}$

Lauren Shelton
Lauren Shelton
Numerade Educator
03:16

Problem 101

The number of terms that are not free from radicals in the expansion of $\left(\mathrm{x}^{1 / 4}+\mathrm{y}^{1 / 20}\right)^{64}$ is
(a) 5
(b) 17
(c) 60
(d) 61

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:48

Problem 102

In a circus there are 15 animals that have to be put in cages 15 in a row, one in each cage. Of the 15 cages available, two are too small for 12 of the animals. Of the remaining 13 cages, 5 are small for 7 of the animals. The number of ways of caging the animals, is
(a) $\frac{8 ! 6 !}{3 !}$
(b) $\frac{8 ! 3 !}{6 !}$
(c) $\frac{8 !}{3 ! 6 !}$
(d) $8 ! 3 ! 6 !$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:49

Problem 103

A large family has 15 children including 2 sets of identical twins, 3 sets of identical triplets and two others. The number of ways of seating these children in a row if the two sets of identical twins have to occupy two extreme seats together is
(a) $\frac{11 !}{3 !}$
(b) $\frac{11 !}{(3 !)^{3}}$
(c) $\frac{2 \times 11 !}{(3 !)^{3}}$
(d) $\frac{11 ! \times 2}{3(3 !)}$

Clarissa Noh
Clarissa Noh
Numerade Educator
03:19

Problem 104

The number of divisors of 453600 which are of the form $8 n+4(n \in N)$ is
(a) 30
(b) 90
(c) 120
(d) 180

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:39

Problem 105

The maximum number of points of intersection of a system of $\mathrm{n}$ circles and $\mathrm{m}$ triangles, given that no two triangles have a common point and no three circles intersect in the same 2 points is
(a) $\mathrm{n}(\mathrm{n}-1)$
(b) $\mathrm{n}(\mathrm{n}-3 \mathrm{~m}+1)$
(c) $n(n+3-1)$
(d) $n(n+6 m-1)$

Carson Merrill
Carson Merrill
Numerade Educator
08:15

Problem 106

Anand and Bimal are first semester B Tech students of IIT in city $\mathrm{X}$. They had their schooling in two different schools located in cities $Y$ and $Z$. Anand has $x$ friends in city $Y$. (who studied with him in the same school). Bimal has $(\mathrm{x}+3)$ friends in city $\mathrm{Z}$ (who studied in the same school with him). In the IIT where they study now, they have common friends. Altogether, there are 12 who are friends of Anand or of Bimal. One day Anand and Bimal arranged a party for their friends and organized some games. Assume that no game is played between two friends who have studied/studying in the same institution. The maximum number of games that could be played, is
(a) 23
(b) 66
(c) 46
(d) 45

Gus Steppen
Gus Steppen
Numerade Educator
02:31

Problem 107

The number of 5-digit numbers divisible by 9 that can be formed using the eight digits 0 to 7 , without repetition is
(a) 240
(b) 384
(c) 624
(d) 264

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:55

Problem 108

If $\frac{{ }^{n} P_{r}}{120}=\frac{{ }^{n} C_{r}}{5}$, then $r$ is
(a) 4
(b) 5
(c) 24
(d) 600

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:53

Problem 109

Number of ways 8 persons can be selected from a group of 11 so as to include 2 particular persons and exclude 1 person is
(a) $11 \mathrm{C} 8-2$
(b) 28
(c) 55
(d) 45

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:28

Problem 110

The exponent of 5 in $120 !$ is
(a) 24
(b) 3
(c) 29
(d) 28

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:38

Problem 111

Statement 1
12 books of different authors are divided into 3 groups each containing 4 books. The number of ways in which this an be done is $\frac{12 !}{(4 !)^{3} \times 3 !}$
Statement 2
Number of ways of selecting $r$ things out of $n$ different things is ${ }^{n} C_{r}$.

AG
Ankit Gupta
Numerade Educator
01:29

Problem 112

Statement 1
Number of positive integral solutions of the equation $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=20$ is 171 and
Statement 2
Number of non-negative integral solutions of $x_{1}+x_{2}+\ldots .+x_{r}=n$ is $^{n+r-1} C_{r-1}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:18

Problem 113

Statement 1
Number of terms in the expansion of $(3 a+5 b-9 c)^{25}$ is 351 .
and
Statement 2
Number of terms in the expansion of $(\mathrm{x}+\mathrm{a})^{\mathrm{n}}$ is $(\mathrm{n}+1)$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:01

Problem 114

The sum of the series $1+\frac{3}{2^{3}}+\frac{1.3}{1.2}\left(\frac{3^{2}}{2^{6}}\right)+\frac{1.3 .5}{1.2 .3}\left(\frac{3^{3}}{2^{9}}\right) \ldots . .$ is a root of the equation $3 x^{2}-a x+6=0 .$
Then a equals
(a) 9
(b) $-9$
(c) 2
(d) $-2$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:06

Problem 115

If $\mathrm{y}=\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3 .5}{4.8 .12} \ldots . .$, then $\mathrm{y}^{2}+2 \mathrm{y}-3$ is
(a) 2
(b) 0
(c) $-2$
(d) $-4$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:24

Problem 116

${ }^{4} \mathrm{C}_{1}+{ }^{5} \mathrm{C}_{2} \frac{1}{2}+{ }^{6} \mathrm{C}_{3}\left(\frac{1}{2}\right)^{2}+{ }^{7} \mathrm{C}_{4}\left(\frac{1}{2}\right)^{3}+{ }^{8} \mathrm{C}_{5}\left(\frac{1}{2}\right)^{4}$.... sums up to the product of the roots of the equation
(a) $x^{3}-10 x^{2}+31 x-30=0$
(b) $x^{3}+10 x^{2}+31 x+30=0$
(c) $x^{3}-10 x^{2}-31 x+30=0$
(d) $2 x^{3}+10 x^{2}-31 x-30=0$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:22

Problem 117

If ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}=36,{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=84$ and ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}+1}=126$, then,
(a) $n=9$
(b) $\quad r=3$
(c) ${ }^{n} P_{r+2}=15120$
(d) $\quad{ }^{(2 \mathrm{n}+\mathrm{r})} \mathrm{C}_{3}=630$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:13

Problem 118

Words are formed using all the letters in the word "MULTIPLE". Then,
(a) Number of words in which the positions of the respective vowels should be as in the word MULTIPLE, is 60
(b) Number of words in which the relative portions of vowels and consonants are not changed, is 180
(c) Number of words in which the order of the vowels occurring in MULTIPLE are not altered, is 360
(d) Number of words starting with $\mathrm{L}$ and ending with $\mathrm{L}$ is 720

Mir Afzal
Mir Afzal
Numerade Educator
01:25

Problem 119

If $\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots \ldots . ., \mathrm{C}_{10}$ are the coefficients of the powers of $\mathrm{x}$ in the expansion of $(1+\mathrm{x})^{10}$
(a) $\mathrm{C}_{0}+2 \times \mathrm{C}_{1}+3 \times \mathrm{C}_{2}+\ldots \ldots . .+11 \times \mathrm{C}_{10}$ equals $2^{9} \times 12$
(b) $\mathrm{C}_{0}^{2}+\mathrm{C}_{1}^{2}+\mathrm{C}_{2}^{2}+\ldots \ldots+\mathrm{C}_{10}^{2}$ equals $\frac{20 !}{(10 !)^{2}}$
(c) $\mathrm{C}_{0}-\mathrm{C}_{1}+\mathrm{C}_{2}-\mathrm{C}_{3}+\mathrm{C}_{4}-+\ldots \ldots+\mathrm{C}_{10}$ equals zero
(d) $\frac{C_{0}}{1}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots \ldots+\frac{C_{10}}{11}$ equals $\frac{2^{11}}{11}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:23

Problem 120

${ }^{n} C_{0},{ }^{n} C_{1} \ldots . .$ are represented by $C_{0}, C_{1} \ldots \ldots \ldots$
Column I
(a) $\mathrm{C}_{0}-2 \mathrm{C}_{1}+3 \mathrm{C}_{2}-4 \mathrm{C}_{3} \cdots \ldots+11 \mathrm{C}_{10}$
(b) $2 C_{0}-3 C_{1}+4 C_{2}-5 C_{3} \cdots+12 C_{10}$
(c) $\frac{C_{0}}{1}-\frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4} \ldots .+\frac{C_{10}}{11}$
(d) $\frac{C_{0}}{2}-\frac{C_{1}}{3}+\frac{C_{2}}{4}-\frac{C_{3}}{5} \ldots . \frac{C_{10}}{12}$
Column II
(p) $\frac{1}{10}$
(q) $\frac{1}{132}$
(r) 0
(s) $\frac{1}{11}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
00:47

Problem 121

Find the number of ways in which two different numbers between 1 and 50 be chosen so that the difference between them is at most 10 .

Aymara Gallardo
Aymara Gallardo
Numerade Educator
01:13

Problem 122

In how many ways can 5 Mathematics, 5 Physics and 5 Chemistry books (all distinct) be arranged in a circular shelf if no two books of the same subjects are together?

Kyler Gray
Kyler Gray
Numerade Educator
01:59

Problem 123

Find the number of numbers greater than $10^{9}$ that can be formed using all the digits of 9440213134 .

James Yang
James Yang
Numerade Educator
01:41

Problem 124

Prove that $2993^{1998}-2956^{1998}-2939^{1998}+2902^{1998}$ is divisible by the LCM of 37 and 54 .

Aman Gupta
Aman Gupta
Numerade Educator
01:49

Problem 125

If $\left(1+x^{3}\right)^{2}(1+x)^{n}=\sum_{k=0}^{n+6} a_{k} x^{k}$ and if $a_{1}, a_{2}, a_{3}$ are in G.P., find the value of $n$.

mp
Manik Pulyani
Numerade Educator
01:57

Problem 126

Find the coefficient of $x^{50}$ in the polynomial obtained after simplifying $(1+x)^{2}+2(1+x)^{3}+\ldots+1000(1+x)^{1001} .$

Akash Goyal
Akash Goyal
Numerade Educator
05:15

Problem 127

Prove that the coefficient of $x^{137}$ in $\left(1-2 x+x^{2}-2 x^{3}+\ldots-2 x^{299}+x^{300}\right) \times\left(1+2 x+x^{2}-2 x^{3}+\ldots+2 x^{299}+x^{300}\right)$ is zero.

Vg
Viraj Gaggar
Numerade Educator
03:30

Problem 128

If $C_{r}={ }^{n} C_{p}$ evaluate $C_{0} C_{2}+C_{1} C_{3}+C_{2} C_{4}+\ldots+C_{n-2} C_{n}$.

Aman Gupta
Aman Gupta
Numerade Educator
01:28

Problem 129

How many pairs $\left(\mathrm{n}_{1}, \mathrm{n}_{2}\right)$ of 4 -digit numbers can be formed so that the digits of $\mathrm{n}_{1}$ are not less than the corresponding digits of $\mathrm{n}_{2}$.

Shahab Ullah
Shahab Ullah
Numerade Educator
02:58

Problem 130

Find the number of ways in which 5 numbers in AP can be selected from $1,2,3, \ldots$ n.

Sanchit Jain
Sanchit Jain
Numerade Educator
01:23

Problem 131

If the coefficients of $x, x^{2}$ and $x^{3}$ in the binomial expansion of $(1+x)^{2 n}$ are in AP, then $2 n^{2}$ equals
(a) $9 n-7$
(b) $7-9 n$
(c) 18
(d) 32

Aman Gupta
Aman Gupta
Numerade Educator
05:30

Problem 132

If $\left(1-x+4 x^{2}\right)^{30}=a_{0}+a_{1} x+\ldots \ldots+a_{60} x^{60}$, then $a_{0}+a_{2}+a_{4}+\ldots+a_{58}$ is
(a) $2^{29}\left(2^{30}+3^{30}\right)$
(b) $2^{29}\left(3^{30}-2^{30}\right)$
(c) $2^{30}\left(2^{30}+3^{30}\right)$
(d) $2^{30}\left(3^{30}-2^{30}\right)$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:44

Problem 133

If $x$ is positive and less than 1 , the first negative term in the expansion of $(1+x)^{27 / 5}$ is
(a) 8 th term
(b) 6 th term
(c) 5 th term
(d) 7 th term

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
06:24

Problem 134

If $\mathrm{p}=\left(1+{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}+{ }^{\mathrm{n}} \mathrm{C}_{3}+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}\right)^{2}$ and $\mathrm{q}=1+{ }^{2 \mathrm{n}} \mathrm{C}_{1}+{ }^{2 \mathrm{n}} \mathrm{C}_{2}+{ }^{2 \mathrm{n}} \mathrm{C}_{3}+\ldots+{ }^{2 \mathrm{n}} \mathrm{C}_{2 \mathrm{n}}$, then
(a) $\mathrm{p}=\mathrm{q}$
(b) $\mathrm{p}=\mathrm{q}^{2}$
(c) $\mathrm{p}^{2}=\mathrm{q}$
(d) $\mathrm{p}=2 \mathrm{q}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:32

Problem 135

10 persons are to be arranged around a round table. 3 persons wish to sit as a group. Number of ways the arrangements can be made, is
(a) $9 ! \times 3 !$
(b) $8 ! \times 3 !$
(c) $7 ! \times 8 \mathrm{P}_{3}$
(d) $7 ! \times 3 !$

Aman Gupta
Aman Gupta
Numerade Educator
02:40

Problem 136

The number of ways in which 5 letters can be put in 5 addressed envelopes such that not a single letter goes into its correctly addressed envelope is
(a) 120
(b) 115
(c) 44
(d) 45

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
02:00

Problem 137

If the 2 nd and 3 rd terms in the expansion of $(x-y)^{n}$ are in the same ratio as the 3 rd and the 4 th terms in the expansion of $(x-y)^{n+3}$, then $n$ equals
(a) 4
(b) 5
(c) 6
(d) 8

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:43

Problem 138

$\sqrt{3}\left\{(\sqrt{3}+1)^{20}-(\sqrt{3}-1)^{20}\right\}$ is
(a) a natural number
(b) a fraction
(c) a negative number
(d) an irrational number

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:54

Problem 139

If $x$ lies between $-1$ and 1 , the coefficient of $x^{6}$ in the expansion of $\left(x^{2}+x+1\right)^{-3}$ is
(a) 6
(b) 9
(c) 3
(d) 4

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:08

Problem 140

There are $\mathrm{n}$ seats around a circular table. If $\mathrm{k}$ and $\ell$ denote respectively the number of ways of seating $\mathrm{m}$ men $(\mathrm{m}<\mathrm{n})$ around this table when the seats are numbered and not numbered, then $\frac{\mathrm{k}}{\ell}$ equals
(a) $\mathrm{n}$
(b) $\mathrm{m}$
(c) $\frac{1}{\mathrm{~m}}$
(d) $\frac{1}{n}$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:40

Problem 141

There are 8 candidates participating in a debate. The number of ways, in which the eight of them could be lined up so that $\mathrm{A}_{2}$ speaks before $\mathrm{A}_{3}$, who speaks before $\mathrm{A}_{4}$ and who speaks before $\mathrm{A}_{5}$, is
(a) $\frac{{ }^{8} \mathrm{P}_{4}}{4 !}$
(b) ${ }^{8} \mathrm{C}_{4} \times 4 !$
(c) $8 ! 4 !$
(d) $\frac{{ }^{8} \mathrm{C}_{4}}{4 !}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:16

Problem 142

The greatest number formed on the screen of a calculator is $99999999 .$ The value of $\mathrm{k}$, where $\mathrm{k}$ denotes the number of whole numbers formed on the screen, which can be recognized as numbers with correct digits when seen as mirror reflection, is
(a) 312501
(b) 312500
(c) 321501
(d) 325101

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
03:26

Problem 143

If the coefficient of $\mathrm{x}^{-1}$ in the expansion of $\left(\mathrm{px}^{3}-\frac{1}{\mathrm{q} \mathrm{x}^{2}}\right)^{8}$ is $\frac{-7}{4}$ and that of $\mathrm{x}^{19}$ is $-4$, then the positive values of $\mathrm{p}$ and q are respectively
(a) 2,1
(b) 1,2
(c) 2,2
(d) 1,1

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:39

Problem 144

If $\sum_{\mathrm{r}=0}^{20} \mathrm{a}_{\mathrm{r}}(\mathrm{x}-4)^{\mathrm{r}}=\sum_{\mathrm{r}=0}^{20} \mathrm{~b}_{\mathrm{r}}(\mathrm{x}-6)^{\mathrm{r}}$ and if $\mathrm{a}_{\mathrm{k}}=1$, for all $\mathrm{k} \geq 10$, then $\mathrm{b}_{19}$ is
(a) $-41$
(b) $-21$
(c) 21
(d) 41

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:10

Problem 145

If $\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}$ be the coefficients of any four consecutive terms in the expansion of $(1+\mathrm{t})^{\mathrm{n}}$, then $\left(\frac{p}{p+q}+\frac{q}{q+r}-\frac{r}{r+s}\right)\left(\frac{q}{q+r}+\frac{r}{r+s}-\frac{p}{p+q}\right)-\frac{4 p r}{(p+q)(r+s)}+\frac{3 q^{2}}{(q+r)^{2}}$
(a) 0
(b) 1
(c) $-1$
(d) 2

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:30

Problem 146

The number of ways in which a selection of four letters can be made from the letters of the word 'PROPORTION' is
(a) 58
(b) 38
(c) 53
(d) 43

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:29

Problem 147

If $\mathrm{N}>\mathrm{n}>1$, then the value of $\sum_{\lambda=0}^{\mathrm{N}}{ }^{\lambda+\mathrm{n}-1} \mathrm{C}_{\mathrm{n}-1}$ is equal to
(a) 1
(b) ${ }^{\mathrm{N}} \mathrm{C}_{\mathrm{n}+1}$
(c) $\mathrm{N}+\mathrm{n} \mathrm{C}$
(d) ${ }^{N+n-1} C_{n-1}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:10

Problem 148

Through each of the angular points of a triangle 75 straight lines are drawn, and no two of the 225 straight lines are parallel, and also no three, one from each angular point meet in a point. Then the number of points of intersection is
(a) ${ }^{225} \mathrm{C}_{2}$
(b) ${ }^{225} \mathrm{C}_{2}-\left({ }^{75} \mathrm{C}_{2}-1\right)$
(c) ${ }^{225} \mathrm{C}_{2}-3\left({ }^{75} \mathrm{C}_{2}\right)$
(d) ${ }^{225} \mathrm{C}_{2}-3\left({ }^{75} \mathrm{C}_{2}-1\right)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:34

Problem 149

If $\mathrm{n}$ belongs to the set of natural numbers then that $2^{3 \mathrm{n}+3}-7 \mathrm{n}-8$ is divisible by
(a) 49
(b) 43
(c) 51
(d) 47

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:20

Problem 150

The coefficient of $x^{100}$ in the expression $(1+x)^{500}+2 x(1+x)^{499}+3 x^{2}(1+x)^{498}+\ldots+501 x^{500}$ is
(a) 499
(b) ${ }^{502} \mathrm{C}_{100}$
(c) ${ }^{500} \mathrm{C}_{100}$
(d) ${ }^{502} \mathrm{C}_{100}-1-502$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:24

Problem 151

If I is the integral part and $\mathrm{F}$ the fractional part of $(3 \sqrt{3}+5)^{2 \mathrm{n}+1}$, where $\mathrm{n}$ is a positive integer, then $\mathrm{F} \times(\mathrm{I}+\mathrm{F})=$
(a) 1
(b) 2
(c) $2^{2 n}$
(d) $2^{2 n+1}$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:10

Problem 152

If $n$ is a positive integer, the integral part of $(4+\sqrt{10})^{n}$ is
(a) an even number
(b) an odd number
(c) zero
(d) cannot be formed

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
04:06

Problem 153

Let $C_{0}, C_{1}, C_{2}, C_{3}, \ldots, C_{n}$ represent the coefficients of the binomial expansion $(1+x)^{n}$ where $n$ is a positive integer. Then $2 \times C_{0}+5 \times C_{1}+8 \times C_{2}+\ldots .+(3 n+2) \times C_{n}=$
(a) 0
(b) $2^{n-1}$
(c) $3 \mathrm{n} 2^{\mathrm{n}-1}$
(d) $(3 n+4) 2^{n-1}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:06

Problem 154

If $C_{0}, C_{1}, C_{2} \ldots$ are binomial coefficient in expansion of $(1+x)^{n}$ then $2 \times C_{0}-3 \times C_{1}+4 \times C_{2}-5 \times C_{3}+\ldots .$ up to $(n+1)$ terms $=$
(a) $(\mathrm{n}+2) 2^{\mathrm{n}}$
(b) $\mathrm{n} 2^{\mathrm{n}-1}$
(c) $\mathrm{n} !$
(d) 0

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:45

Problem 155

If $C_{0}, C_{1}, C_{2} \ldots C_{n}$ are binomial coefficients of order $n$, then $\sum_{0 \leq i_{i}, \leq n, i \neq j} C_{i} C_{j}=$
(a) $2^{2 \mathrm{n}}-\mathrm{n} !$
(b) $2^{2 n-1}-\frac{(2 n) !}{2(n !)^{2}}$
(c) $2^{2 n-1}-(2 n) !$
(d) 0

Aman Gupta
Aman Gupta
Numerade Educator
04:06

Problem 156

If $\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \mathrm{C}_{3}, \ldots, \mathrm{C}_{\mathrm{n}}$ are the coefficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$ where $\mathrm{n}$ is a odd positive integer, then $C_{0}^{2}-C_{1}^{2}+C_{2}^{2}-C_{3}^{2}+-\ldots+(-1)^{n} C_{n}^{2}=$
(a) 0
(b) $\frac{(2 n) !}{2 !}$
(c) $\mathrm{n} !$
(d) 1

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:59

Problem 157

The number of positive integral solutions of the equation $x_{1}+2 x_{2}+x_{3}+x_{4}=18$ is
(a) 78
(b) 305
(c) 308
(d) 385

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
04:36

Problem 158

The number of non-negative integral solutions of the equation $\mathrm{x}_{1}-\mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4}=4 ; 0 \leq \mathrm{x}_{\mathrm{i}} \leq 4, \mathrm{i}=1,2,3,4, \ldots$ is
(a) 85
(b) 80
(c) 75
(d) 63

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
02:18

Problem 159

There are some men and 4 women participating in a tournament. Every participant plays four games with every other participant. If the total number of games played between men participants exceeds that played between men and women participants by 380 , then the number of men participants is
(a) 16
(b) 17
(c) 19
(d) 21

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:31

Problem 160

Through each of the angular points of a triangle $m$ straight lines are drawn and no two of the $3 \mathrm{~m}$ lines are parallel; also no three, one from each angular point, meet at a point. Then the number of points of intersection. (Other than angular points of the triangle)
(a) $3 \mathrm{~m}$
(b) $3 \mathrm{~m}^{2}-3$
(c) $3 \mathrm{~m}^{2}$
(d) $\mathrm{m}^{2}$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:42

Problem 161

If three successive coefficients in the expansion of $\left(x^{2}+\frac{p}{x}\right)$ are in the ratio $21: 35: 40$ and if the coefficient of the
second term is 20 , then the values of $n$ and $p$ are respectively
(a) 10,2
(b) 8,4
(c) 2,8
(d) 12,2

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:59

Problem 162

The number of positive integral solutions of $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=50$
(a) 276
(b) 1176
(c) 576
(d) 572

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:12

Problem 163

There are $\mathrm{n}$ straight lines in a plane, no two of which are parallel and no three of them are concurrent. Triangles are formed with their vertices at the points of intersection of these straight lines. The number of such triangles formed is
(a) ${ }^{\mathrm{n}} \mathrm{C}_{3}-{ }^{(\mathrm{n}-1)} \mathrm{C}_{3}$
(b) ${ }^{\left(\mathrm{n} \mathrm{C}_{2}\right)} \mathrm{C}_{3}-\mathrm{n}\left({ }^{(\mathrm{n}-1)} \mathrm{C}_{3}\right)$
(c) ${ }^{\mathrm{n}} \mathrm{C}_{3}+{ }^{(\mathrm{n}-1)} \mathrm{C}_{3}$
(d) ${ }^{n} C_{3}-n\left({ }^{(n-1)} C_{3}\right)$

Evan Sun
Evan Sun
Numerade Educator
01:13

Problem 164

Let $(x, y, z, t)$ be selected from the set $\{1,2,3, \ldots \ldots \ldots, n\} .$ The number of 4 -tuples $(x, y, z, t)$ where $x \leq y \leq z \leq t$ and not all $\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t}$ are equal, is
(a) ${ }^{\mathrm{n}} \mathrm{C}_{4}+{ }^{\mathrm{n}} \mathrm{C}_{3}+{ }^{\mathrm{n}} \mathrm{C}_{2}$
(b) ${ }^{(\mathrm{n}+2)} \mathrm{C}$
(c) ${ }^{(\mathrm{n}+3)} \mathrm{C}_{4}$
(d) ${ }^{(n+3)} C_{4}-n$

Grant Castaneda
Grant Castaneda
Numerade Educator
02:32

Problem 165

The number of different 10 letter codes that can be formed using the characters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ with the restriction that $\mathrm{A}$ appears exactly thrice and $\mathrm{B}$ appears exactly twice in each such code is
(a) 60840
(b) 88400
(c) 80640
(d) $\quad 64080$

James Yang
James Yang
Numerade Educator
03:15

Problem 166

If in the expansion of $\left(\frac{7}{2}+\frac{x}{7}\right)^{n}, n \in N$ the 7 th term is the greatest when $x=\frac{1}{2}$ then the number of possible values for $n$ is
(a) 7
(b) 49
(c) 43
(d) 14

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:40

Problem 167

The sum of odd factors of $10 !$ that are of the form $5 \mathrm{~m}+2, \mathrm{~m} \in \mathrm{N}$ is
(a) 34
(b) 567
(c) 601
(d) 594

Aman Gupta
Aman Gupta
Numerade Educator
01:20

Problem 168

The number of four digit numbers having exactly 3 consecutive digits identical, is
(a) 162
(b) 153
(c) 90
(d) 163

Aman Gupta
Aman Gupta
Numerade Educator
01:40

Problem 169

The number of odd proper divisors of $3^{\mathrm{m}} \times 6^{\mathrm{n}} \times 21^{\mathrm{p}}$ is
(a) $(m+1)(n+1)(p+1)-1$
(b) $(m+n+p+1)(p+1)-1$
(c) $(m+1)(n+1)(p+1)-2$
(d) $(m+1)(n+1)(p+1)$

mp
Manik Pulyani
Numerade Educator
02:23

Problem 170

The number of ways an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question is
(a) ${ }^{21} \mathrm{C}_{7}$
(b) ${ }^{21} \mathrm{C}_{8}$
(c) ${ }^{21} \mathrm{C}_{9}$
(d) ${ }^{22} \mathrm{C}_{7}$

Cheyenne Whinham
Cheyenne Whinham
Numerade Educator
04:45

Problem 171

Statement 1 $25 \times 26 \times 27 \times \ldots . . \times 150$ is divisible by $126 !$
and
Statement 2 ${ }^{\mathrm{n}} \mathrm{C}$ is a positive integer.

Shaza Hammoud
Shaza Hammoud
Numerade Educator
04:06

Problem 172

$\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots$ are the coefficients of powers of $\mathrm{x}$ in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$ where $\mathrm{n}$ is a positive integer. Statement 1 If $n=20, \frac{C_{1}}{C_{0}}+2 \times \frac{C_{2}}{C_{1}}+3 \times \frac{C_{3}}{C_{2}}+\ldots . .+20 \times \frac{C_{20}}{C_{19}}=210$ and Statement 2 $\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}+\ldots .+\mathrm{C}_{\mathrm{n}}=2^{\mathrm{n}}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:45

Problem 173

Statement 1 $1 !+2 !+3 !+4 !+\ldots .+12 !$ when divided by 7 leaves remainder 5. and
Statement 2 Product of $\mathrm{N}$ consecutive positive integers is divisible by $\mathrm{N}$ !

AG
Ankit Gupta
Numerade Educator
02:28

Problem 174

Statement 1
$$
\left|\begin{array}{lll}
{ }^{10} \mathrm{C}_{6} & { }^{10} \mathrm{C}_{7} & { }^{11} \mathrm{C}_{7} \\
{ }^{12} \mathrm{C}_{5} & { }^{12} \mathrm{C}_{6} & { }^{13} \mathrm{C}_{6} \\
{ }^{14} \mathrm{C}_{7} & { }^{14} \mathrm{C}_{8} & { }^{15} \mathrm{C}_{8}
\end{array}\right|=0
$$
and
Statement 2
${ }^{n} C_{r-1}+{ }^{n} C_{r}={ }^{n+1} C_{r}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:42

Problem 175

Statement 1 and
Statement 2 The term independent of $\mathrm{x}$ is the expansion of $\left(\mathrm{x}^{2}+\frac{\mathrm{b}}{\mathrm{x}}\right)^{\mathrm{n}}$ where, $\mathrm{n}$ is an even integer is the middle term.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:24

Problem 176

Statement 1 The number of ways of selecting 3 coins from a bag containing 5 five rupee coins, 5 two rupee coins and 5 one rupee coins is ${ }^{15} \mathrm{C}$ and Statement 2 Number of selection of $\mathrm{r}$ things from $\mathrm{n}$ distinct things is ${ }^{\mathrm{n}} \mathrm{C}$.

Kumar Vaibhav
Kumar Vaibhav
Numerade Educator
02:18

Problem 177

Statement 1 Number of distinct terms in the expansion $(a x+b y+c z)^{20}$ is 231 . and
Statement 2 Number of distinct terms in the expansion of $(a x+b y)^{n}$ is $(n+1)$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:29

Problem 178

Statement 1 5 balls of different colours are to be put in 3 distinct boxes so that no box is empty. Number of ways is $150 .$ and Statement 2 Number of positive integral solutions of $\mathrm{x}+\mathrm{y}+\mathrm{z}=5$ is 6

Aman Gupta
Aman Gupta
Numerade Educator
01:46

Problem 179

Statement 1 Number of rational terms in the expansion of $\left(\sqrt{2}+\frac{1}{5 \sqrt{2}}\right)^{25}$ is 12 . and
Statement 2 $(r+1)$ th term in the expansion of $(a+b)^{n}$ is ${ }^{n} C_{r} a^{n-r} b^{r}$

Aman Gupta
Aman Gupta
Numerade Educator
05:30

Problem 180

Consider the expansion $(1+x)^{12}=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots \ldots+a_{12} x^{12}$
Statement 1 $\mathrm{a}_{0}-\mathrm{a}_{2}+\mathrm{a}_{4}-\mathrm{a}_{6}+-\ldots \ldots+\mathrm{a}_{12}=-64$ and Statement 2 $\mathrm{a}_{0}+\mathrm{a}_{2}+\mathrm{a}_{4}+\mathrm{a}_{6}+\mathrm{a}_{8}+\mathrm{a}_{10}+\mathrm{a}_{12}=2^{11}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:35

Problem 181

The number of even positive integers that could be formed using the digits $0,1,2,3,4$ and 5, not using any digit more than once in any number is given by
(a) $5 !$
(b) $5(6 !)$
(c) 847
(d) $6(5 !)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
06:32

Problem 182

3-digit numbers are formed using the digits $0,2,3,5$ and 6 without repetition. The number of ways of getting a number which is divisible by 4 or 6 is
(a) 16
(b) 8
(c) $4 !$
(d) $3 !$

Bharat Bairagi
Bharat Bairagi
Numerade Educator
02:31

Problem 183

A 10 -digit number is formed using all the digits $0,1,2,3 \ldots \ldots .9$ without repetition. The number of numbers divisible
by 36 is
(a) $9 \times 9 !$
(b) $20 \times 7 !$
(c) $20 \times 8 !$
(d) $5 \times 7 !$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:26

Problem 184

The number of ways of selecting 2 numbers $x$ and $y$ from the set of the first 50 natural numbers such that $x^{2}-y^{2}$ is divisible by 5 is
(a) 2500
(b) 425
(c) 225
(d) 400

Aman Gupta
Aman Gupta
Numerade Educator
02:31

Problem 185

The number of numbers formed using the digits $3,4,5,6,7,8,9$ not more than once, divisible by 225 is
(a) $4 !$
(b) 33
(c) $7 !$
(d) 5!

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
07:35

Problem 186

The number of triads of 3 integers, having their sum divisible by 5, chosen from the first 600 natural numbers is
(a) $\quad{ }^{120} \mathrm{C}_{3}+2 \times\left({ }^{120} \mathrm{C}_{1}\right)^{3}+4\left({ }^{120} \mathrm{C}_{2}\right)\left({ }^{120} \mathrm{C}_{1}\right)$
(b) $\left({ }^{120} \mathrm{C}_{3}\right)^{3}$
(c) ${ }^{120} \mathrm{C}_{1}+{ }^{120} \mathrm{C}_{2}+{ }^{120} \mathrm{C}_{3}$
(d) $\left({ }^{120} \mathrm{C}_{3}\right)^{2}+{ }^{120} \mathrm{C}_{1}$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:25

Problem 187

Number of non-negative integral solutions of the equation $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=8$ is
(a) 459
(b) 495
(c) 395
(d) 359

Aman Gupta
Aman Gupta
Numerade Educator
01:59

Problem 188

Number of non-negative integral solutions of the equation $x_{1}+2 x_{2}+x_{3}=18$ is
(a) 100
(b) 50
(c) 75
(d) 105
189. Number of positive integral solutions of the equation $\mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}=22$ is
(a) 60
(b) 62
(c) 36
(d) 63

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:59

Problem 189

Number of positive integral solutions of the equation $\mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}=22$ is
(a) 60
(b) 62
(c) 36
(d) 63

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:12

Problem 190

If $\mathrm{n}$ is a positive integer greater than $1,(1+\mathrm{x})^{\mathrm{n}}-\mathrm{n} \mathrm{x}-1$ is divisible by
(a) $\mathrm{x}$
(b) $\mathrm{x}^{2}$
(c) $x^{3}$
(d) $x^{4}$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
10:14

Problem 191

Consider the integer and between 1 and $10^{5}$
(a) Number of integer and which contain exactly one 4 is 32805
(b) Number of integer and which contain exactly one 4 , one 5 and one 7 is 2940
(c) Number of integer and which contain exactly one 4 , one 5 and one 7 is 3230
(d) Number of integer and which contain exactly one 5 and one 9 is 12912

Anthony Ramos
Anthony Ramos
Numerade Educator
03:11

Problem 192

Let $(1+x)^{10}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots . .+C_{10} x^{10}$, where $C_{r}={ }^{10} C_{r}, r=0,1,2 \ldots ., 10 .$ Then, which of the following results hold good?
(a) $\mathrm{C}_{1}+2 \times \mathrm{C}_{2}+3 \times \mathrm{C}_{3}+4 \times \mathrm{C}_{4}+\ldots .+10 \times \mathrm{C}_{10}=5120$
(b) $\frac{C_{0}}{1}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots \ldots+\frac{C_{10}}{11}=\frac{2047}{11}$
(c) $\mathrm{C}_{0}+\left(\mathrm{C}_{0}+\mathrm{C}_{1}\right)+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}\right)+\ldots .+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}++\mathrm{C}_{10}\right)=6144$
(d) $\mathrm{C}_{1}+\mathrm{C}_{3}+\mathrm{C}_{5}+\mathrm{C}_{7}=512$

Kristen Marte
Kristen Marte
Numerade Educator
03:57

Problem 193

Number of ways of making 3 parcels with 5 identical maths books and 5 identical chemistry books so that each of the three parcels contains at least one maths book and one chemistry book, is equal to
(a) 36
(b) $5^{6}$
(c) $\mathrm{k}^{2}$, where $\mathrm{k}$ denotes the number of positive integral solutions of the equation $\mathrm{x}+\mathrm{y}+\mathrm{z}=5$
(d) the number of positive integral solution of $x+y+z=10$

Clarissa Noh
Clarissa Noh
Numerade Educator
04:44

Problem 194

The number of selections of 4 letters taken from the word ASSISTANCE is
(a) 72
(b) $\frac{1}{6} \times{ }^{13} \mathrm{C}_{2}$
(c) ${ }^{7} \mathrm{C}_{3}+2{ }^{7} \mathrm{C}_{2}-5$
(d) 48

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
01:27

Problem 195

The number of arrangements of the integer $1,2,3,4,5,6,7,8$ taken all at a time such that the product of any two consecutive integers is even, is
(a) ${ }^{5} \mathrm{P}_{4} \times 4 !$
(b) $(4 !)^{2}$
(c) $4 !$
(d) 1152

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
06:08

Problem 196

${ }^{24} \mathrm{C}_{\mathrm{r}}+2 \times{ }^{24} \mathrm{C}_{\mathrm{r}+1}+{ }^{24} \mathrm{C}_{\mathrm{r}+2}>{ }^{26} \mathrm{C}_{10}$ when $\mathrm{r}=$
(a) 9
(b) 10
(c) 11
(d) 12

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:03

Problem 197

Let $\mathrm{n}$ be a positive integer and $\left(1+3 \mathrm{x}+3 \mathrm{x}^{2}+\mathrm{x}^{3}\right)^{\mathrm{n}}=\sum_{\mathrm{r}=0}^{3 \mathrm{n}} \mathrm{A}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}}$ Then,
(a) $\mathrm{A}_{\mathrm{r}}=\mathrm{A}_{3 \mathrm{n}-\mathrm{r}}, \mathrm{r}=0,1,2, \ldots ., 3 \mathrm{n}$
(b) $A_{r}=A_{n-r}, r=0,1,2, \ldots,, n$
(c) $\mathrm{A}_{0}+\mathrm{A}_{1}+\mathrm{A}_{2}+\ldots \ldots+\mathrm{A}_{3 \mathrm{n}}=8^{\mathrm{n}}$
(d) $\mathrm{A}_{0}-\mathrm{A}_{1}+\mathrm{A}_{2}-+\ldots \ldots .+(-1)^{3 \mathrm{n}} \mathrm{A}_{3 \mathrm{n}}=1$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:26

Problem 198

Constant term in the expansions of
Column I $\quad$ Column II
(a) $\left(3 x^{2}-\frac{2}{x}\right)^{15}$
(p) ${ }^{15} \mathrm{C}_{4} \times 18^{4} \times 7$
(b) $(1+x)^{8}\left(1+\frac{1}{x}\right)^{8}$
(q) ${ }^{15} \mathrm{C}_{4} \frac{2^{4} .71}{15.3^{10}}$
(c) $\left(x+\frac{1}{x}\right)^{3}\left(3 x+\frac{2}{x^{2}}\right)^{15}$
(r) ${ }^{15} \mathrm{C}_{4}\left(\frac{11}{5}\right)(12)^{5}$
(d) $\left(1+x+x^{2}+x^{3}+x^{4}+x^{5}\right)\left(2 x^{2}+\frac{1}{3 x}\right)^{15}$
(s) ${ }^{15} \mathrm{C}_{4}\left(\frac{11}{7}\right)(2 \times 3)$

Nick Johnson
Nick Johnson
Numerade Educator
03:42

Problem 199

Column I
(a) The number of positive integers which can be formed using the digits 0 . $1,2,4$ where each of these digits is used at most once.
(b) There are 10 stations on a circular railway network. A train has to stop at 3 stations such that no two stations are adjacent. The number of such selections is
(c) All the students of a class send new year greetings to one another. If the postmen deliver 2352 greeting cards to students of this class the number of students in the class is
(d) If $\mathrm{N}$ denotes the maximum number of points of intersection of 4 circles and 4 straight lines, $\mathrm{N}$ equals
Column II
(p) 48
(q) 49
(r) 50
(s) 51

Carson Merrill
Carson Merrill
Numerade Educator
03:51

Problem 200

Column I
(a) If $n$ is the number of terms in the expansion of $(3 x+y-5 z+t)^{30}, \frac{n}{31}$ equals
(b) If $^{\mathrm{n}-1} \mathrm{C}_{14}+{ }^{\mathrm{n}-1} \mathrm{C}_{15}>{ }^{\mathrm{n}} \mathrm{C}_{13}$, then $\mathrm{n}$ equals
(c) If $\mathrm{N}$ is the number of 11 digit numbers that can be formed from the number 33338888662 by rearranging the digits so that odd digits occupy even places, $\frac{\mathrm{N}}{25}$ equals $=$
(d) Number of integral solutions of the equation $x_{1}+x_{2}+x_{3}=25$, where $5 \leq x_{i} \leq 10$, $\mathrm{i}=1,2,3$ is
Column II
(p) 21
(q) 176
(r) 30
(s) 144

Ahmad Reda
Ahmad Reda
Numerade Educator