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,

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