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Introductory Combinatorics

Richard A. Brualdi

Chapter 5

The Binomial Coefficients - all with Video Answers

Educators

Chapter Questions

01:14

Problem 1

Prove Pascal's formula by substituting the values of the binomial coefficients as given in equation (5.1).

AG
Ankit Gupta
Numerade Educator
03:15

Problem 2

Fill in the rows of Pascal's triangle corresponding to $n=9$ and $10 .$

Angela Guo
Angela Guo
Numerade Educator
02:34

Problem 3

Consider the sum of the binomial coefficients along the diagonals of Pascal's triangle running upward from the left. The first few are $1,1,1+1=2,1+2=$ $3,1+3+1=5,1+4+3=8$. Compute several more of these diagonal sums, and determine how these sums are related. (Compare them with the values of the counting function $f$ in Exercise 4 of Chapter $1 .$ )

Anna Jones
Anna Jones
Numerade Educator
02:29

Problem 4

Expand $(x+y)^{5}$ and $(x+y)^{6}$ using the binomial theorem.

AG
Ankit Gupta
Numerade Educator
01:46

Problem 5

Expand $(2 x-y)^{7}$ using the binomial theorem.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:00

Problem 6

What is the coefficient of $x^{5} y^{13}$ in the expansion of $(3 x-2 y)^{18} ?$ What is the coefficient of $x^{8} y^{9} ?$ (There is not a misprint in this last question!)

Aymara Gallardo
Aymara Gallardo
Numerade Educator
02:06

Problem 7

Use the binomial theorem to prove that
$$
3^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) 2^{k} .
$$
Generalize to find the sum
$$
\sum_{k=0}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) r^{k}
$$
for any real number $r .$

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:06

Problem 8

Use the binomial theorem to prove that
$$
2^{n}=\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}
n \\
k
\end{array}\right) 3^{n-k} .
$$

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:31

Problem 9

Evaluate the sum
$$
\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}
n \\
k
\end{array}\right) 10^{k} .
$$

AG
Ankit Gupta
Numerade Educator
00:40

Problem 10

Use combinatorial reasoning to prove the identity (5.2).

Hunza Gilgit
Hunza Gilgit
Numerade Educator
03:55

Problem 11

Use combinatorial reasoning to prove the identity (in the form given)
$$
\left(\begin{array}{l}
n \\
k
\end{array}\right)-\left(\begin{array}{c}
n-3 \\
k
\end{array}\right)=\left(\begin{array}{l}
n-1 \\
k-1
\end{array}\right)+\left(\begin{array}{l}
n-2 \\
k-1
\end{array}\right)+\left(\begin{array}{l}
n-3 \\
k-1
\end{array}\right) .
$$
(Hint: Let $S$ be a set with three distinguished elements $a, b$, and $c$ and count certain $k$ -subsets of $S$.)

Aymara Gallardo
Aymara Gallardo
Numerade Educator
01:09

Problem 12

Let $n$ be a positive integer. Prove that
$$
\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}
n \\
k
\end{array}\right)^{2}=\left\{\begin{array}{ll}
0 & \text { if } n \text { is odd } \\
(-1)^{m}\left(\begin{array}{c}
2 m \\
m
\end{array}\right) & \text { if } n=2 m .
\end{array}\right.
$$
(Hint: For $n=2 m$, consider the coefficient of $x^{n}$ in $\left(1-x^{2}\right)^{n}=(1+x)^{n}(1-x)^{n}$.)

Carson Merrill
Carson Merrill
Numerade Educator
01:18

Problem 13

Find one binomial coefficient equal to the following expression:
$$
\left(\begin{array}{l}
n \\
k
\end{array}\right)+3\left(\begin{array}{c}
n \\
k-1
\end{array}\right)+3\left(\begin{array}{c}
n \\
k-2
\end{array}\right)+\left(\begin{array}{c}
n \\
k-3
\end{array}\right) .
$$

David Nguyen
David Nguyen
Numerade Educator
03:07

Problem 14

Prove that
$$
\left(\begin{array}{l}
r \\
k
\end{array}\right)=\frac{r}{r-k}\left(\begin{array}{c}
r-1 \\
k
\end{array}\right)
$$
for $r$ a real number and $k$ an integer with $r \neq k$.

Kevin Harmer
Kevin Harmer
Numerade Educator
02:46

Problem 15

Prove, that for every integer $n>1$,
$$
\left(\begin{array}{l}
n \\
1
\end{array}\right)-2\left(\begin{array}{l}
n \\
2
\end{array}\right)+3\left(\begin{array}{l}
n \\
3
\end{array}\right)+\cdots+(-1)^{n-1} n\left(\begin{array}{l}
n \\
n
\end{array}\right)=0 .
$$

Sanchit Jain
Sanchit Jain
Numerade Educator
01:44

Problem 16

By integrating the binomial expansion, prove that, for a positive integer $n$,
$$
1+\frac{1}{2}\left(\begin{array}{l}
n \\
1
\end{array}\right)+\frac{1}{3}\left(\begin{array}{l}
n \\
2
\end{array}\right)+\cdots+\frac{1}{n+1}\left(\begin{array}{l}
n \\
n
\end{array}\right)=\frac{2^{n+1}-1}{n+1}
$$.

Sherin Hussain
Sherin Hussain
Numerade Educator
01:46

Problem 17

Prove the identity in the previous exercise by using (5.2) and (5.3).

Mahendra Kumar
Mahendra Kumar
Numerade Educator
02:32

Problem 18

Evaluate the sum
$$
1-\frac{1}{2}\left(\begin{array}{l}
n \\
1
\end{array}\right)+\frac{1}{3}\left(\begin{array}{l}
n \\
2
\end{array}\right)-\frac{1}{4}\left(\begin{array}{c}
n \\
3
\end{array}\right)+\cdots+(-1)^{n} \frac{1}{n+1}\left(\begin{array}{l}
n \\
n
\end{array}\right) .
$$

Mir Afzal
Mir Afzal
Numerade Educator
09:01

Problem 19

Sum the series $1^{2}+2^{2}+3^{2}+\cdots+n^{2}$ by observing that
$$
m^{2}=2\left(\begin{array}{c}
m \\
2
\end{array}\right)+\left(\begin{array}{c}
m \\
1
\end{array}\right)
$$
and using the identity (5.19).

Sandip Ranjan
Sandip Ranjan
Numerade Educator
05:28

Problem 20

Find integers $a, b$, and $c$ such that
$$
m^{3}=a\left(\begin{array}{c}
m \\
3
\end{array}\right)+b\left(\begin{array}{c}
m \\
2
\end{array}\right)+c\left(\begin{array}{c}
m \\
1
\end{array}\right)
$$
for all $m$. Then sum the series $1^{3}+2^{3}+3^{3}+\cdots+n^{3}$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:07

Problem 21

Prove that, for all real numbers $r$ and all integers $k$,
$$
\left(\begin{array}{c}
-r \\
k
\end{array}\right)=(-1)^{k}\left(\begin{array}{c}
r+k-1 \\
k
\end{array}\right) .
$$

Kevin Harmer
Kevin Harmer
Numerade Educator
03:07

Problem 22

Prove that, for all real numbers $r$ and all integers $k$ and $m$,
$$
\left(\begin{array}{c}
r \\
m
\end{array}\right)\left(\begin{array}{l}
m \\
k
\end{array}\right)=\left(\begin{array}{l}
r \\
k
\end{array}\right)\left(\begin{array}{c}
r-k \\
m-k
\end{array}\right) .
$$

Kevin Harmer
Kevin Harmer
Numerade Educator
00:59

Problem 23

Every day a student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She always takes a shortest walk of 24 blocks.
(a) How many different walks are possible?
(b) Suppose that four blocks east and five blocks north of her home lives her best friend, whom she meets each day on her way to school. Now how many different walks are possible?
(c) Suppose, in addition, that three blocks east and six blocks north of her friend's house there is a park where the two girls stop each day to rest and play. Now how many different walks are there?
(d) Stopping at a park to rest and play, the two students often get to school late. To avoid the temptation of the park, our two students decide never to pass the intersection where the park is. Now how many different walks are there?

Ashley Volpe
Ashley Volpe
Numerade Educator
01:13

Problem 24

Consider a three-dimensional grid whose dimensions are 10 by 15 by 20. You are at the front lower left corner of the grid and wish to get to the back upper right corner 45 "blocks" away, How many different routes are there in which you walk exactly 45 blocks?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
03:15

Problem 25

Use a combinatorial argument to prove the Vandermonde convolution for the binomial coefficients: For all positive integers $m_{1}, m_{2}$, and $n$,
$$
\sum_{k=0}^{n}\left(\begin{array}{c}
m_{1} \\
k
\end{array}\right)\left(\begin{array}{c}
m_{2} \\
n-k
\end{array}\right)=\left(\begin{array}{c}
m_{1}+m_{2} \\
n
\end{array}\right)
$$
Deduce the identity $(5.16)$ as a special case.

Clarissa Noh
Clarissa Noh
Numerade Educator
04:25

Problem 26

Let $n$ and $k$ be integers with $1 \leq k \leq n$. Prove that
$$
\sum_{k=1}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right)\left(\begin{array}{c}
n \\
k-1
\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}
2 n+1 \\
n+1
\end{array}\right)-\left(\begin{array}{c}
2 n \\
n
\end{array}\right)
$$.

Aymara Gallardo
Aymara Gallardo
Numerade Educator
03:15

Problem 27

Let $n$ and $k$ be positive integers. Give a combinatorial proof of the identity $(5.15)$
$$
n(n+1) 2^{n-2}=\sum_{k=1}^{n} k^{2}\left(\begin{array}{l}
n \\
k
\end{array}\right) .
$$

Clarissa Noh
Clarissa Noh
Numerade Educator
05:24

Problem 28

Let $n$ and $k$ be positive integers. Give a combinatorial proof that
$$
\sum_{k=1}^{n} k\left(\begin{array}{l}
n \\
k
\end{array}\right)^{2}=n\left(\begin{array}{c}
2 n-1 \\
n-1
\end{array}\right)
$$.

Aymara Gallardo
Aymara Gallardo
Numerade Educator
02:59

Problem 29

Find and prove a formula for
$$
\sum_{r, s, t \geq 0}\left(\begin{array}{c}
m_{1} \\
r
\end{array}\right)\left(\begin{array}{c}
m_{2} \\
s
\end{array}\right)\left(\begin{array}{c}
m_{3} \\
t
\end{array}\right)
$$
where the summation extends over all nonnegative integers $r, s$ and $t$ with sum $r+s+t=n$.

Supratim Pal
Supratim Pal
Numerade Educator
01:30

Problem 30

Prove that the only antichain of $S=\{1,2,3,4\}$ of size 6 is the antichain of all 2-subsets of $S$.

Nick Johnson
Nick Johnson
Numerade Educator
04:30

Problem 31

Prove that there are only two antichains of $S=\{1,2,3,4,5\}$ of size 10 (10 is maximum by Sperner's theorem), namely, the antichain of all 2-subsets of $S$ and the antichain of all 3 -subsets.

JW
Julian Wong
Numerade Educator
04:30

Problem 32

- Let $S$ be a set of $n$ elements. Prove that, if $n$ is even, the only antichain of size $\left(\begin{array}{l}\mathrm{n} \\ \left.\frac{n}{2}\right]\end{array}\right)$ is the antichain of all $\frac{n}{2}$ -subsets; if $n$ is odd, prove that the only antichains of this size are the antichain of all $\frac{n-1}{2}$ -subsets and the antichain of all $\frac{n+1}{2}$ -subsets.

JW
Julian Wong
Numerade Educator
04:59

Problem 33

Construct a partition of the subsets of $\{1,2,3,4,5\}$ into symmetric chains.

Heather Zimmers
Heather Zimmers
Numerade Educator
01:00

Problem 34

In a partition of the subsets of $\{1,2, \ldots, n\}$ into symmetric chains, how many chains have only one subset in them? two subsets? $k$ subsets?

Rikhil Makwana
Rikhil Makwana
Numerade Educator
01:45

Problem 35

A talk show host has just bought 10 new jokes. Each night he tells some of the jokes. What is the largest number of nights on which you can tune in so that you never hear on one night at least all the jokes you heard on one of the other nights? (Thus, for instance, it is acceptable that you hear jokes 1,2, and 3 on one night, jokes 3 and 4 on another, and jokes 1,2 , and 4 on a third. It is not acceptable that you hear jokes 1 and 2 on one night and joke 2 on another night.)

Shu Naito
Shu Naito
Numerade Educator
03:10

Problem 36

Prove the identity of Exercise 25 using the binomial theorem and the relation $(1+x)^{m_{1}}(1+x)^{m_{2}}=(1+x)^{m_{1}+m_{2}} .$

Kamalesh Bagrecha
Kamalesh Bagrecha
Numerade Educator
06:48

Problem 37

Use the multinomial theorem to show that, for positive integers $n$ and $t$.
$$
t^{n}=\sum\left(\begin{array}{c}
n \\
n_{1} n_{2} \cdots n_{t}
\end{array}\right)
$$,
where the summation extends over all nonnegative integral solutions $n_{1}, n_{2}, \ldots, n_{t}$ of $n_{1}+n_{2}+\cdots+n_{t}=n$.

Chris Trentman
Chris Trentman
Numerade Educator
02:40

Problem 38

Use the multinomial theorem to expand $\left(x_{1}+x_{2}+x_{3}\right)^{4}$.

Rylie Howey
Rylie Howey
Numerade Educator
03:58

Problem 39

Determine the coefficient of $x_{1}^{3} x_{2} x_{3}^{4} x_{5}^{2}$ in the expansion of
$$
\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)^{10}
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:11

Problem 40

What is the coefficient of $x_{1}^{3} x_{2}^{3} x_{3} x_{4}^{2}$ in the expansion of
$$
\left(x_{1}-x_{2}+2 x_{3}-2 x_{4}\right)^{9} ?
$$

Linh Vu
Linh Vu
Numerade Educator
03:19

Problem 41

Expand $\left(x_{1}+x_{2}+x_{3}\right)^{n}$ by observing that
$$
\left(x_{1}+x_{2}+x_{3}\right)^{n}=\left(\left(x_{1}+x_{2}\right)+x_{3}\right)^{n}
$$
and then using the binomial theorem.

Joanna Quigley
Joanna Quigley
Numerade Educator
03:30

Problem 42

Prove the identity (5.21) by a combinatorial argument. (Hint: Consider the permutations of a multiset of objects of $t$ different types with repetition numbers $n_{1}, n_{2}, \ldots, n_{t}$, respectively. Partition these permutations according to what type of object is in the first position.)

WZ
Wen Zheng
Numerade Educator
07:24

Problem 43

Prove by induction on $n$ that, for $n$ a positive integer,
$$
\frac{1}{(1-z)^{n}}=\sum_{k=0}^{\infty}\left(\begin{array}{c}
n+k-1 \\
k
\end{array}\right) z^{k}, \quad|z|<1 .
$$
Assume the validity of
$$
\frac{1}{1-z}=\sum_{k=0}^{\infty} z^{k}, \quad|z|<1 .
$$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
05:52

Problem 44

Prove that
$$
\sum_{n_{1}+n_{2}+n_{3}=n}\left(\begin{array}{cc}
n & \\
n_{1} n_{2} n_{3}
\end{array}\right)(-1)^{n_{1}-n_{2}+n_{3}}=1
$$
where the summation extends over all nonnegative integral solutions of $n_{1}+n_{2}+$ $n_{3}=n .$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
05:52

Problem 45

Prove that
$$
\sum_{n_{1}+n_{2}+n_{3}+n_{4}=n}\left(\begin{array}{cc}
n & \\
n_{1} n_{2} n_{3} n_{4}
\end{array}\right)(-1)^{n_{1}-n_{2}+n_{3}-n_{4}}=0
$$
where the summation extends over all nonnegative integral solutions of $n_{1}+n_{2}+$ $n_{3}+n_{4}=n .$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:03

Problem 46

Use Newton's binomial theorem to approximate $\sqrt{30}$.

Angela Guo
Angela Guo
Numerade Educator
01:32

Problem 47

Use Newton's binomial theorem to approximate $10^{1 / 3}$.

AG
Ankit Gupta
Numerade Educator
03:36

Problem 48

Use Theorem $5.6 .1$ to show that, if $m$ and $n$ are positive integers, then a partially ordered set of $m n+1$ elements has a chain of size $m+1$ or an antichain of size $n+1$

Willis James
Willis James
Numerade Educator
02:51

Problem 49

Use the result of the previous exercise to show that a sequence of $m n+1$ real numbers either contains an increasing subsequence of $m+1$ numbers or a decreasing subsequence of $n+1$ numbers (see Application 9 of Section $2.2$ ).

Lauren Shelton
Lauren Shelton
Numerade Educator
11:24

Problem 50

Consider the partially ordered set $(X, \mid)$ on the set $X=\{1,2, \ldots, 12\}$ of the first 12 positive integers, partially ordered by "is divisible by."
(a) Determine a chain of largest size and a partition of $X$ into the smallest. number of antichains.
(b) Determine an antichain of largest size and a partition of $X$ into the smallest number of chains.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
08:12

Problem 51

Let $R$ and $S$ be two partial orders on the same set $X$. Considering $R$ and $S$ as subsets of $X \times X$, we assume that $R \subseteq S$ but $R \neq S$. Show that there exists an ordered pair $(p, q)$, where $(p, q) \in S$ and $(p, q) \notin R$ such that $R^{\prime}=R \cup\{(p, q)\}$ is also a partial order on $X$. Show by example that not every such $(p, q)$ has the property that $R^{\prime}$ is a partial order on $X$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator