Yes, I didn't explain very clearly. In your example, what I'm saying is that I'd like to prove that the number of ways in which we can find non-negative integers (also, previously I said "positive integers" - I meant non-negative integers) such that
is 28 possibilities.
(I said " " last time - I meant . Sorry again!)
Just for clarity, I'll list them here:
(0,0,0,0,0,0), (1,0,0,0,0,0), (1,0,0,0,0,1), (1,0,0,0,1,0), (1,0,0,1,0,0), (1,0,1,0,0,0), (1,1,0,0,0,0), (0,1,0,0,0,0), (0,1,0,0,0,1), (0,1,0,0,1,0), (0,1,0,1,0,0), (0,1,1,0,0,0), (0,0,1,0,0,0), (0,0,1,0,0,1), (0,0,1,0,1,0), (0,0,1,1,0,0), (0,0,0,1,0,0), (0,0,0,1,0,1), (0,0,0,1,1,0), (0,0,0,0,1,0), (0,0,0,0,1,1), (0,0,0,0,0,1), (2,0,0,0,0,0), (0,2,0,0,0,0), (0,0,2,0,0,0), (0,0,0,2,0,0), (0,0,0,0,2,0), (0,0,0,0,0,2).
So there are indeed 28. I'd like a way of proving that the number of ways will be in the general case for any t and n.
It's easy to prove the above by induction but I agree that there looks like there should be some nicer, intuitive combinatorial argument.
I'm struggling with the part before though - how do we know that the number of ways of finding non-negative integers such that (for a given positive integer k)
is ? Sorry if I'm overlooking something obvious but have been thinking about this for a while.