Chapter 15.7, Problem 27WE

To determine

To prove:

The value of the given number.

Explanation of Solution

Given:

The order of operation

  ${}^n{C}_r={}^n{C}_{n-r}$

Concept used:

Using by the formula of the permutation

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

Calculation:

The order of operation

  ${}^n{C}_r={}^n{C}_{n-r}\mathrm{......................}\left(1\right)$

Using by the formula in equation (1)

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

Therefore,

The value of given operation is $\frac{n!}{r!\left(n-r\right)!}$ .