Answer
The required solution is $362880$
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 $_{9}{{P}_{9}}$.
Here, $ n=9,r=9$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{9}{{P}_{9}}=\frac{9!}{\left( 9-9 \right)!} \\
& =\frac{9!}{0!} \\
& =\frac{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{0!}
\end{align}$
Since, $0!=1$.
Therefore, $\begin{align}
& \frac{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{0!}=\frac{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{1} \\
& =\frac{362880}{1} \\
& =362880
\end{align}$
Thus, $_{9}{{P}_{9}}=362880$
