Answer
The required solution is $6720$
Work Step by Step
We know that the representation $_{n}{{P}_{r}}$ implies that the number of possible well-organized arrangements of n items is taken r at a time.
And the number of possible well-organized arrangements of n items taken r at a time can be evaluated as:
$_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
And the provided expression is $_{8}{{P}_{5}}$.
Here, $ n=8,r=5$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{8}{{P}_{5}}=\frac{8!}{\left( 8-5 \right)!} \\
& =\frac{8!}{3!} \\
& =\frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3!}{3!}
\end{align}$
And simplify further, $\begin{align}
& \frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3!}{3!}=8\cdot 7\cdot 6\cdot 5\cdot 4 \\
& =6720
\end{align}$
Hence, $_{8}{{P}_{5}}=6720$
