College Algebra, Chapter 9, Review Exercises, Section Review Exercises, Problem 74

Evaluate the expression
$
\displaystyle
\sum^{8}_{k = 0}
\left(
\begin{array}{c}
8\\
k
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-k
\end{array}
\right)
$



$
\begin{equation}
\begin{aligned}
\sum^{8}_{k = 0}
\left(
\begin{array}{c}
8\\
k
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-k
\end{array}
\right)
&=

\left(
\begin{array}{c}
8\\
0
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-0
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
1
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-1
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
2
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-2
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
3
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-3
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
4
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-4
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
5
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-5
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
6
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-6
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
7
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-7
\end{array}
\right)

+

\left(
\begin{array}{c}
8\\
8
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-8
\end{array}
\right)
\end{aligned}
\end{equation}
$

Recall for the 8th row of the Pascal's Triangle...

$
\left(
\begin{array}{c}
8\\
0
\end{array}
\right)

= 1,

\left(
\begin{array}{c}
8\\
1
\end{array}
\right)

= 8,

\left(
\begin{array}{c}
8\\
2
\end{array}
\right)

= 28,

\left(
\begin{array}{c}
8\\
3
\end{array}
\right)

= 56,

\left(
\begin{array}{c}
8\\
4
\end{array}
\right)

= 70,
$



$
\left(
\begin{array}{c}
8\\
5
\end{array}
\right)

= 56,

\left(
\begin{array}{c}
8\\
6
\end{array}
\right)

= 28,

\left(
\begin{array}{c}
8\\
7
\end{array}
\right)

= 8,

\left(
\begin{array}{c}
8\\
8
\end{array}
\right)

= 1

$

Thus,

$
\begin{equation}
\begin{aligned}
\sum^{8}_{k = 0}

\left(
\begin{array}{c}
8\\
k
\end{array}
\right)
\left(
\begin{array}{c}
8\\
8-k
\end{array}
\right)
&=
(1)(1) + (8)(8) + (28)(28) + (56)(56) + (70)(70) + (56)(56) + (28)(28) + (8)(8) + (1)(1)\\
\\
&= 12870
\end{aligned}
\end{equation}
$