Combinations.

Let $\large N=\prod_{i=1}^{k}\binom{n-\sum_{j=1}^{i-1}r_{j}}{r_{i}}$ Where $\large\sum_{i=1}^{k}r_{i}\leq n$

Since N is a product of Binomial coefficients, it is a positive integer. But, $\large N=\prod_{i=1}^{k}\frac{(n-\sum_{j=1}^{i-1}r_{j})!}{(r_{i}!)(n-\sum_{j=1}^{i}r_{j})!}$ $\large =\frac{n!}{r_{1}!(n-r_{1})!}\times\frac{(n-r_{1})!}{r_{2}!(n-r_{1}-r_{2})!}........$ $\large =\frac{n!}{r_{1}!r_{2}!....r_{k}!(n-\sum_{j=1}^{k}r_{j})!}$ Therefore, The expression given in the question is $\large N\times(n-\sum_{j=1}^{k}r_{j})!$ Which is an integer.