Chapter 9, Problem 1RE

Solution

To calculate: The value of the expression $\left(\begin{array}{c}12\\ 5\end{array}\right)$ .

The resultant value is $\left(\begin{array}{c}12\\ 5\end{array}\right)=792$ .

Given information:

The given expression is $\left(\begin{array}{c}12\\ 5\end{array}\right)$ .

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

Calculation:

Consider the given expression

  $\left(\begin{array}{c}12\\ 5\end{array}\right)$

Rewrite the expression using the formula $\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$ ,

  $\begin{array}{c}\left(\begin{array}{c}12\\ 5\end{array}\right)=\frac{12!}{5!\left(12-5\right)!}\\ =\frac{12!}{5!7!}\end{array}$

Now expand the factorial,

  $\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7!}{\left(5\cdot 4\cdot 3\cdot 2\cdot 1\right)7!}=792$

Therefore, the value of $\left(\begin{array}{c}12\\ 5\end{array}\right)$ is 792.