College Algebra, Chapter 9, 9.6, Section 9.6, Problem 18

Evaluate the expression
$\left( \begin{array}{c} 8\\ 3 \end{array} \right)$

Recall that the binomial coefficient is denoted by $\displaystyle \left( \frac{n}{r} \right)$ and is defined by
$\displaystyle \left( \begin{array}{c} n\\ r \end{array} \right) = \frac{n!}{r!(n-r)!}$
where $n$ and $r$ are non negative numbers with $r \leq n$
Substituting $n = 8$ and $r = 3$ gives

$\begin{equation} \begin{aligned} \begin{array}{c} 8\\ 3 \end{array} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} &= \frac{8 \cdot 7 \cdot 6 \cdot \cancel{5\cdot4\cdot3\cdot2\cdot1}}{(3\cdot2\cdot1)(\cancel{5\cdot4\cdot3\cdot2\cdot1})}\\ \\ &= \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1}\\ \\ &= \frac{336}{6}\\ \\ &= 56 \end{aligned} \end{equation}$