Chapter 8, Problem 1SE

To determine

To evaluate: The number $C\left(9,7\right)$.

Answer to Problem 1SE

The value of the number $C\left(9,7\right)$ is $\underset{\_}{36}$.

Explanation of Solution

Formula used:

Number of different r-combination:

The number of different r-combinations of a set of n distinct elements is $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$.

Calculation:

By using the above mentioned formula, find the value of the number $C\left(9,7\right)$ by substituting $n=9$ and $r=7$ as follows.

$\begin{array}{c}C\left(9,7\right)=\frac{9!}{7!\left(9-7\right)!}\\ =\frac{9!}{7!2!}\\ =\frac{9\cdot 8\cdot 7!}{7!2!}\\ =\frac{9\cdot 8}{2}\end{array}$

On further simplification,

$\begin{array}{c}C\left(9,7\right)=9\cdot 4\\ =36\end{array}$

Therefore, the value of the number $C\left(9,7\right)$ is $\underset{\_}{36}$.