One way to do it is to note that is the coefficient of in the binomial . Since the coefficients of x and y are 1, all such coefficients are positive integers.
And, of course, with n= a+ b and k= b.
If you don't like that, try proving it by induction on b with a fixed.
If b= 1, , an integer.
Assume true for given b and look at .
I'll try a "Proof By Induction" on this,
though it has not been as straightforward as the PBI proofs I've done up to now.
Aside from that, the following is an attempted proof using PBI.
and prove it must be a positive integer for all of the terms being positive integers,
then deduce the binomial from that by setting all but 2 terms to zero,
Or, we can simply work with the binomial exclusively.
So, taking the multinomial coefficient...
This means that the integers summing to k may be different sets of integers, but they do sum to k.
It also means that is an integer variable, that is, it can vary but takes on only positive integer values.
Therefore, the base case involves establishing this framework for initial values.
For example, suppose k=5 (though we would start with k=smallest practical value of interest)...
for positive integers, and n=2 or 3.
The set of integers sum to k+1.
However, we utilise the fact that
Hence the inductive step is complete.