Answer
$12,870$
Work Step by Step
$n!=n(n-1)(n-2)\cdot...\cdot 2\cdot 1$
$0!=1$
$\displaystyle \left(\begin{array}{l}
n\\
r
\end{array}\right)=\frac{n!}{r!(n-r)!}$
Property: $\left(\begin{array}{l}
n\\
r
\end{array}\right)=\left(\begin{array}{l}
n\\
n-r
\end{array}\right)$
-------------------------
$\left(\begin{array}{l}
8\\
k
\end{array}\right)\left(\begin{array}{l}
8\\
8-k
\end{array}\right)=\left(\begin{array}{l}
8\\
k
\end{array}\right)^{2}$
sum=$\displaystyle \sum_{k=0}^{8} \left(\begin{array}{l}
8\\
k
\end{array}\right)^{2}$
$ \left(\begin{array}{l}
8\\
0
\end{array}\right)=1= \left(\begin{array}{l}
8\\
8
\end{array}\right)$
$ \left(\begin{array}{l}
8\\
1
\end{array}\right)=8= \left(\begin{array}{l}
8\\
7
\end{array}\right)$
$ \displaystyle \left(\begin{array}{l}
8\\
2
\end{array}\right)=\frac{8(7)}{1(2)}=28= \left(\begin{array}{l}
8\\
6
\end{array}\right)$
$ \displaystyle \left(\begin{array}{l}
8\\
3
\end{array}\right)=\frac{8(7)(6)}{1(2)(3)}=56= \left(\begin{array}{l}
8\\
5
\end{array}\right)$
$ \displaystyle \left(\begin{array}{l}
8\\
4
\end{array}\right)=\frac{8(7)(6)(5)}{1(2)(3)(4)}=70$
$=2(1)+2\cdot 8^{2}+2\cdot(28)^{2}+2\cdot(56)^{2}+(70)^{2}$
$=12,870$
