College Algebra, Chapter 9, 9.6, Section 9.6, Problem 22

Evaluate the expression
$
\left(
\begin{array}{c}
5\\
2
\end{array}
\right)
\left(
\begin{array}{c}
5\\
3
\end{array}
\right)
$

Recall that the binomial coefficient is denoted by $\displaystyle \left( \frac{n}{r} \right)$ and is defined by
Substituting $n = 5$ and $r = 2$ gives

$
\begin{equation}
\begin{aligned}
\begin{array}{c}
5\\
2
\end{array}
=
\frac{5!}{2!(5-2)!}
=
\frac{5!}{2!3!}
&=
\frac{5\cdot 4 \cdot \cancel{3 \cdot 2 \cdot 1}}{(2\cdot1)(\cancel{3 \cdot 2 \cdot 1})}\\
\\
&= \frac{5 \cdot 4}{2 \cdot 1} = \frac{20}{2}\\
\\
&= 10
\end{aligned}
\end{equation}
$


Substituting $n = 5$ and $r = 3$ gives

$
\begin{equation}
\begin{aligned}
\left(
\begin{array}{c}
5\\
3
\end{array}
\right)
=
\frac{5!}{3!(5-3)!}
=
\frac{5!}{3!2!}
&=
\frac{5\cdot 4 \cdot \cancel{3 \cdot 2 \cdot 1}}{\cancel{3 \cdot 2 \cdot 1}(2\cdot 1)}\\
\\
&= \frac{5 \cdot 4}{2 \cdot 1} = \frac{20}{2}\\
\\
&= 10
\end{aligned}
\end{equation}
$

Thus,

$
\left(
\begin{array}{c}
5\\
2
\end{array}
\right)
\left(
\begin{array}{c}
5\\
3
\end{array}
\right)
= (10)(10)
= 100
$