For non-negative integers $n$ and $k$ the binomial coefficient $\binom{n}k$ is the number of $k$-sized subsets of a set of $n$ things: it’s the number of different combinations of $k$ of the $n$ elements. The members of a $k$-sized set can be listed in $k!$ different orders, so there are altogether $k!\binom{n}k$ permutations of $k$-sized subsets of the original set of $n$ things, $k!$ orderings of each of $\binom{n}k$ sets.
But $\dbinom{n}k=\dfrac{n!}{k!(n-k)!}$, so $k!\binom{n}k=k!\cdot\frac{n!}{k!(n-k)!}=\frac{n!}{(n-k)!}=n(n-1)(n-2)\dots(n-k+1)\;,$
and you often see the formula for the number of permutations of $k$-sized subsets of an $n$-set expressed in one of these last two ways instead of as $k!\binom{n}k$. In some ways these are simpler than the form $k!\binom{n}k$. However, because the binomial coefficients turn out to be rather easy to work with, and because many relationships involving them are known, it’s often convenient to write the number of permutations as $k!\binom{n}k$ in order to take advantage of those known relationships.