Answer
The required solution is $792$
Work Step by Step
We know that the representation $_{n}{{C}_{r}}$ counts the number of assortments of n items taken r at a time.
And the number of assortment of n items taken r at a time can be evaluated as:
$_{n}{{C}_{r}}=\frac{n!}{\left( n-r \right)!r!}$
And the provided expression is $_{12}{{C}_{5}}$.
Here, $ n=12,r=5$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{12}{{C}_{5}}=\frac{12!}{\left( 12-5 \right)!5!} \\
& =\frac{12!}{7!5!} \\
& =\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7!}{7!\left( 5\cdot 4\cdot 3\cdot 2\cdot 1 \right)}
\end{align}$
And simplify further, $\begin{align}
& \frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7!}{7!\left( 5\cdot 4\cdot 3\cdot 2\cdot 1 \right)}=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{5\cdot 4\cdot 3\cdot 2\cdot 1} \\
& =\frac{95040}{120} \\
& =792
\end{align}$
Hence, $_{12}{{C}_{5}}=792$
