Question

Evaluate the expression.

$C\left(10,3\right)$

Short answer

The value of the expression$C\left(10,3\right)$ is 120.

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 C10, 3.

Use equation (1) to evaluate the expression $C\left(10,3\right)$.

$$\begin{gathered} C\left(10,3\right)=\frac{10!}{\left(10-3\right)!3!} \\ =\frac{10!}{7!3!} \\ =\frac{\left(10\right)\left(9\right)\left(8\right)\left(7!\right)}{7!3!} \\ =\frac{10\times 9\times 8}{3\times 2\times 1} \\ =\frac{720}{6} \\ =120 \end{gathered}$$

Therefore, the value of the expression$C\left(10,3\right)$ is 120.