Answer
The required solution is $720$
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 $_{6}{{P}_{6}}$.
Here, $ n=6,r=6$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{6}{{P}_{6}}=\frac{6!}{\left( 6-6 \right)!} \\
& =\frac{6!}{0!} \\
& =\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{0!}
\end{align}$
Since, $0!=1$.
Therefore, $\begin{align}
& \frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{0!}=\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{1} \\
& =\frac{720}{1} \\
& =720
\end{align}$
Thus, $_{6}{{P}_{6}}=720$
