Question

Evaluate the expression.

$C\left(9,5\right)$

Short answer

The value of the expression$C\left(9,5\right)$ is 126.

Step-by-step solution

Step 1. Define combination.

A selection of objects in which order is not important is called a combination.

Step 2. Define the meaning of Cn, r.

The number of combinations of r objects taken from the group of n objects is given by

$C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}\mathrm{............}\left(1\right)$

Step 3. Evaluate the expression C9, 5.

Use equation (1) to evaluate the expression $C\left(9,5\right)$.

$$\begin{gathered} C\left(9,5\right)=\frac{9!}{\left(9-5\right)!5!} \\ =\frac{9!}{4!5!} \\ =\frac{\left(9\right)\left(8\right)\left(7\right)\left(6\right)\left(5\right)\left(4!\right)}{4!5!} \\ =\frac{9\times 8\times 7\times 6\times 5}{5\times 4\times 3\times 2\times 1} \\ =126 \end{gathered}$$

Therefore, the value of the expression$C\left(9,5\right)$ is 126.