Answer
The required solution is $210$
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:
$_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
The provided expression is $_{7}{{P}_{3}}$.
Here, $ n=7,r=3$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{7}{{P}_{3}}=\frac{7!}{\left( 7-3 \right)!} \\
& =\frac{7!}{4!} \\
& =\frac{7\cdot 6\cdot 5\cdot 4!}{4!}
\end{align}$
And simplify further, $\begin{align}
& \frac{7\cdot 6\cdot 5\cdot 4!}{4!}=7\cdot 6\cdot 5 \\
& =210
\end{align}$
Hence, $_{7}{{P}_{3}}=210$
