While working on a combinatorics/probability question that I had, and in counting dice outcomes for unfair dice I noticed a behavior that I'm struggling a bit to see if it's a generally true property. Rather, I'm fairly certain that it has to be a generally true property, but I'm at a bit of a loss for why.

Let n be a non-negative integer. Let j be one or more positive integers j1, j2, j3, ..., jk such that the sum j1 j2 j3 ... jk = n.

Statement: n! is always evenly divisible by the product of the factorials j1! * j2! * j3! * ... * jk! for any valid j

For example, let n = 6.

So j could be one of:

6
1,5
2,4
3,3
1,1,4
1,2,3
2,2,2
1,1,1,3
1,1,2,2
1,1,1,1,2
1,1,1,1,1,1

If we pick 1,2,3:

6! / (1! * 2! * 3!) = 720 / 12 = 60

The same is true for any j above; I've checked with Python.

Is the same true for any n and j that satisfies the requirements?

Edit: Missed 3,3.