Chapter 12, Problem 8PTTS

To determine

To find: The value of $9{\text{C}}_5$ .

Answer to Problem 8PTTS

The value of $9{\text{C}}_5$ is $126$ .

Explanation of Solution

Given information: $9{\text{C}}_5$ .

Calculation: Here, $9{\text{C}}_5$ represents the number of combinations for sample space $9\text{ choose 5}$ .

The representation of the formula is ${\text{nC}}_r=\frac{\text{n!}}{\text{r!(n}-\text{r)!}}$ where, $\text{n = 9, and r = 5}$

Let’s find the number of combinations from the equation $(1)$ ,

  $9{\text{C}}_5=\frac{9!}{5!(9-5)!}$..............(1)

  $9{\text{C}}_5=\frac{9!}{5!(4)!}\mathrm{......}(2)$

Where,

  $\begin{array}{c}9!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9\\ 5!=1\times 2\times 3\times 4\times 5\\ 4!=1\times 2\times 3\times 4\end{array}$

Therefore, substituting the values of $9!,\text{ 5!, and 4!}$ in the equation $(2)$ ,

  $\begin{array}{l}9{\text{C}}_5=\frac{1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9}{(1\times 2\times 3\times 4\times 5)(1\times 2\times 3\times 4)}\\ 9{\text{C}}_5=\frac{6\times 7\times 8\times 9}{1\times 2\times 3\times 4}\\ 9{\text{C}}_5=\frac{3\times 7\times 8\times 9}{3\times 4}\\ 9{\text{C}}_5=7\times 2\times 9\\ 9{\text{C}}_5=126\end{array}$

Hence, The value of $9{\text{C}}_5$ is $126$ .