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}
$