Chapter Questions
If ${ }^{11} C_{x}={ }^{11} C_{y}$ and $x \neq y$, then what is the value of $x+y$ ?
If ${ }^{18} C_{r}={ }^{18} C_{r+2}$, then find the value of $r$.
If ${ }^{n} C_{7}={ }^{n} C_{3}$, then find value of $n$
If ${ }^{13} C_{r}={ }^{13} C_{r+2}$, then find the value of $r$.
If ${ }^{20} C_{r}={ }^{20} C_{2 r+5}$, then find the value of $r$.
If ${ }^{n} C_{2}=21$, then find value of $n$.
Six persons meet and shake hands. How many hand-shakes are there in all?
Find $n$ and $r$ if(a) ${ }^{n} C_{r}=120,{ }^{n} C_{r}=20$(b) ${ }^{n} C_{r-1}:{ }^{n} C_{r}{ }^{n} C_{e, 1}=2: 3: 4$
If,${ }^{12} C_{5}+2 .{ }^{12} C_{4}+{ }^{12} C_{3}={ }^{14} C_{r}$, then find the value of $r$.
If ${ }^{n} C_{3}+{ }^{n} C_{16}$, find the values of ${ }^{n} C_{23}$ and ${ }^{27} C_{n}$.
Find the value of$${ }^{20} C_{8}+{ }^{20} C_{9}-{ }^{20} C_{11}-{ }^{20} C_{12}$$
If $^{n+2} C_{8}:^{n-2} P_{4}=57: 16$, find the values of $n .$
If ${ }^{n} C_{r}=x .{ }^{n} P_{r}$, find the values of $x$.
If ${ }^{n} P_{r}=504$ and ${ }^{n} C_{r},=84$, find the values of $n$ and $r$.
If $^{n+2}, C_{8}:{ }^{n-2} P_{4}=57: 16$, find the values of $n$.
If ${ }^{22} C_{r}={ }^{22} C_{r+1}$, find the values of ${ }^{r} C_{4}$.
Prove that ${ }^{n} C_{r}+{ }^{n-1} C_{r-1}+{ }^{n-1} C_{r-2}={ }^{n+1} C_{r}$.
If $m={ }^{n} C_{2}$, then prove that ${ }^{m} C_{2}=3 .{ }^{n+1} C_{4}$.If $m={ }^{n} C_{2}$, then prove that ${ }^{m} C_{2}=3 .{ }^{n+1} C_{4}$.
If ${ }^{1000} C_{98}={ }^{999} C_{97}+{ }^{x} C_{901}$, then find $x$.
If ${ }^{13} C_{6}+2 .{ }^{13} C_{5}+{ }^{13} C_{4}={ }^{15} C_{x}$, find the values of $x$.
Prove that$${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$$
There are eight candidates for a particular post. In how many ways can a selection be made of four amongst them, so that(a) 2 persons whose qualifications are below par are included?(b) 2 persons whose qualifications are outstanding are included?
In how many ways can a committee of 5 be formed from 4 professors and 6 students so as to include at least 2 professors?
In an examination of mathematics, 10 questions are set. On how many different ways can a candidate choose 6 questions to answers? If, however, question No.1, be made compulsory, in how many ways can he select questions in all?
A man has 7 friends. In how many ways may he invite one or more of them to a party?
A man has 20 acquaintances of which 12 are relatives. In how many ways out of them he may invite 13 guests so that 9 of them would be relatives and one his best friend?
From 7 commerce and 3 science students in how many ways a committee of 5 students can be formed so as to include at least one science student?
From 6 bowlers, 2 wicket keepers and 8 batsmen, in how many ways a team of 11 players consisting of at least 4 bowlers, one wicket keeper and at least 5 batsmen be formed?
A person has got 15 acquaintances of whom 10 are relatives. In how many ways, may he invite 9 guests so that 7 of them would be relatives?
How many different committees of 5 members can be formed from 6 gentlemen and 4 ladies so as to include (a) 2 ladies, (b) at the most 2 ladies, (c) at least 2 ladies and 2 gentlemen?
At the annual meeting of a society, 7 persons have filed nominations for the post of 5 directors to be elected by the members. Find the number of ways in which a member may choose his vote, if he can vote for not more than the number of post.
Cricket team of 11 is to be selected from a list of 15 players. How many such selections can be made so as to (a) include captain and vice-captain, (b) include a particular player on disciplinary ground?
In a test, a candidate may answer at least one from 5 given questions. Find the different selections he can make.
In an examination paper, there are 7 questions in Group A, out of which any 4 are to be attempted by a candidate and there are 6 questions in Group $\mathrm{B}$, out of which any 3 are to be attempted. In how many different ways can the paper be attempted in full?
From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition, including at least 4 boys and 4 girls. The two girls who won the prize last year must be included. Find the number of ways of such a selection.
How many (a) lines and (b) triangles can be drawn through $n$ points on a circle?
There are 9 consonants and 5 vowels. How many words of 7 letters can be formed using 4 consonants and 3 vowels?
A box contains 12 lamps of which 5 are defective. In how many ways, can a sample of 6 be selected at random from the box so as to include at most 3defective lamps?
A football team of 11 players is to be selected from 2 groups consisting of 8 and 7 players. In how many ways, can the selection be made so that no group contributes less than 5 players?
An examinee is required to answer 6 questions out of 12, which are divided into two groups each containing 6 questions. He is not allowed to answer more than 4 questions from any group. In how many ways, can he answer 6 questions?
A box contains 7 red, 6 white and 4 black balls. How many selections of 3 balls can be made so that (a) all there are red balls, (b) none is red ball and (c) there is one ball of each colour?
In how many different ways, can 9 men be selected from 15 men so as to always (a) include 3 particular men and (b) exclude 3 particular men?
There are 6 bowlers in a cricket team of 14 players. In how many ways, can a team of 11 cricketers be chosen such that at least 4 bowlers are to be included?
A committee of 5 is to be formed from 4 ladies and 6 gentlemen. In how many ways, this can be formed when the committee must contain at least one lady and two gentlemen?
Find the number of permutations of the letters of the words FORECAST and MILKEY, taking 5 at a time so that 3 letters can be chosen from the first word and 2 letters from the second word.
There are 4 fifty-rupee, 5 twenty-rupee and 3 ten-rupee notes. How many selections of one fifty-rupee, one twenty-rupee notes and one ten-rupee notes are possible?
A person has in his bag 11 notes of Rs 100 each, 6 notes of Rs 50 each, 8 notes of Rs 20 each and 4 notes of Rs 10 each. In how many different ways, can he donate a charitable fund?
From 10 candidates, how many selections of 5 can be made so as to (a) include both the youngest and the oldest and (b) exclude the youngest if it includes the oldest?
Find the total number of combinations taking at least one green ball and one blue ball, from 5 different green balls, 4 different blue balls and 3 different red balls.