Answer
The required solution is $5040$
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:
$_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
andthe provided expression is $_{10}{{P}_{4}}$.
Here, $ n=10,r=4$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{10}{{P}_{4}}=\frac{10!}{\left( 10-4 \right)!} \\
& =\frac{10!}{6!} \\
& =\frac{10\cdot 9\cdot 8\cdot 7\cdot 6!}{6!}
\end{align}$
Simplifying further, $\begin{align}
& \frac{10\cdot 9\cdot 8\cdot 7\cdot 6!}{6!}=10\cdot 9\cdot 8\cdot 7 \\
& =5040
\end{align}$
Hence, $_{10}{{P}_{4}}=5040$
