Chapter 9.1, Problem 18E

Solution

To calculate: The value of $C_{{10}_3}$ .

The resultant value is $C_{{10}_3}=120$ .

Given information:

The given expression is $C_{{10}_3}$ .

Formula used: The number of combinations of n objects taken r at a time denoted by $C_{n_r}$ and given by $C_{n_r}=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the given expression

  $C_{{10}_3}$

Rewrite the expression using the formula $C_{n_r}=\frac{n!}{r!\left(n-r\right)!}$ ,

  $\begin{array}{c}{C}_{{10}_3}=\frac{10!}{3!\left(10-3\right)!}\\ =\frac{10!}{3!7!}\end{array}$

Now expand the factorial,

  $\begin{array}{c}\frac{10\cdot 9\cdot 8\cdot 7!}{7!\left(3\cdot 2\cdot 1\right)}=\frac{720}{6}\\ =120\end{array}$

Therefore, the value of $C_{{10}_3}$ is 120.