Is there any known algorithm for factoring the fractional components of a binomial?

Question

For a binomial such as $\binom {15} {6}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}$, it seems that it always divides evenly into an integer, and I understand that there is a proof that this fraction must be an integer. Is there a way other than trial and error to determine which numbers in the denominator divide into which numbers in the numerator?

Answer 1

$\binom{n}{k}=\frac{n!}{k!(n-k)!}$, and we want to show that $k!(n-k)!$ divides $n!$, try to think of $$n!=k!(n-k)!\binom{n}{k}$$ For the left hand side, $n!$ is the number of arrangement for $n$ objects.

For the right hand side, to arrange those $n$ objects, you can first arrange $k$ of them, that gives you $\binom{n}{k}k!$, then arrange rest of the $n-k$, that is $(n-k)!$, thus the total way of arranging $n$ objects is $k!(n-k)!\binom{n}{k}$.

Since the right hand side and the left hand side count the same thing, therefore $k!(n-k)!$ must divide $n!$