Chapter 10.5, Problem 42E

a.

To determine

To prove:

Given identities is true for any whole number $r$ and $n$ .

Answer to Problem 42E

The Expression ${}_n{C}_n=1$ is true for any whole numbers $r$ and $n$ , where $0\le r\le n$ .

Explanation of Solution

Given information:

Condition:

  $0\le r\le n$

Expression:

  ${}_n{C}_n=1$

Formula used:

Expression for combination

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$

Where,

  $n=$ objects

  $r=$ time

Proof:

When $r=n$ , the expression becomes as follows:

  $\begin{array}{l}{}_n{C}_n=\frac{n!}{\left(n-n\right)!\cdot n!}\\ =\frac{n!}{1\cdot n!}\\ =1\end{array}$

For, $n=r=5$ , the expression becomes as follows:

  $\begin{array}{l}{}_5{C}_5=\frac{5!}{\left(5-5\right)!\cdot 5!}\\ =\frac{5!}{1\cdot 5!}\\ =1\end{array}$

Hence, it is proved that ${}_n{C}_n=1$ is true for any whole numbers $r$ and $n$ , where $0\le r\le n$ .

b.

To determine

To prove:

Given identities is true for any whole number $r$ and $n$ .

Answer to Problem 42E

The Expression ${}_n{C}_r{=}_n{C}_{n-r}$ is true for any whole numbers $r$ and $n$ , where $0\le r\le n$ .

Explanation of Solution

Given information:

Condition:

  $0\le r\le n$

Expression:

  ${}_n{C}_r{=}_n{C}_{n-r}$

Formula used:

Expression for permutation

  ${}_n{P}_r{=}_n{C}_r\cdot r!$

Where,

  $n=$ objects

  $r=$ time

Proof:

It is known that:

  ${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substituting the value of ${}_n{P}_r$ in equation ${}_n{P}_r{=}_n{C}_r\cdot r!$ , the value of ${}_n{C}_r$ becomes as follows:

  $\begin{array}{l}{}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}\\ {=}_n{C}_{n-r}\end{array}$

For $n=5,r=2$ , the expression becomes as follows:

  $\begin{array}{l}{}_5{P}_2=\frac{5!}{\left(5-2\right)!}\\ =10\end{array}$

And,

  $\begin{array}{l}{}_5{P}_3=\frac{5!}{3!\left(5-3\right)!}\\ =10\end{array}$

Hence, it is proved that ${}_n{C}_r{=}_n{C}_{n-r}$ is true for any whole numbers $r$ and $n$ , where $0\le r\le n$ .

c.

To determine

To prove:

Given identities is true for any whole number $r$ and $n$ .

Answer to Problem 42E

The expression ${}_{n+1}{C}_r{=}_n{C}_r{+}_n{C}_{r-1}$ is true for any whole numbers $r$ and $n$ , where $0\le r\le n$ .

Explanation of Solution

Given information:

Condition:

  $0\le r\le n$

Expression:

  ${}_{n+1}{C}_r{=}_n{C}_r{+}_n{C}_{r-1}$

Proof:

The given expression is as follows:

  ${}_n{C}_r{+}_n{C}_{r-1}=\left(n+1\right)C_r$

For $n=5,r=2$ , the expression becomes as follows:

  ${}_5{C}_3{+}_5{C}_2=\left(5+1\right)C_3$

Now, ${}_5{C}_3{=}_5{C}_2=10$ and ${}_6{C}_3$ is $20$ .

Therefore,

  $\begin{array}{l}{}_5{C}_3{+}_5{C}_2=\left(5+1\right)C_3\\ 10+10=20\end{array}$

Hence, it is proved that ${}_{n+1}{C}_r{=}_n{C}_r{+}_n{C}_{r-1}$ is true for any whole numbers $r$ and $n$ , where $0\le r\le n$ .