Whenever I solve a problem, I try if I can proceed on with the generalized statement (or expression) for the problem.
I came up with two interesting generalizations today.
- The straight forward one - The number of ordered n-tuples of integers such that
is equal to provided that
- This one generalizes the type of problems where you need to find sum of binomial coefficients which are at certain gaps. By gaps I mean, suppose you need to find as you can find here that binomial coefficients are at certain gaps of . (I hope I am able to explain my point clearly). So here's the generalization - for such that we have
Prove both the generalizations.