Find the value of m for which the given sum is maximum:

$$\sum_{i=0}^m {10\choose i}{20\choose {m-i}}$$ where $${p\choose {q}}=0$$ if p is less than q.

The answer is 15 (given) but how do I find it?

EDIT: Solving it using Vandermonde's identity is fine, but if possible please try to provide a solution without using it.

allows one to sum it into $(10+20) \choose {(i+(m-i))}$=$30\choose {m}$ which is maximal for the center coefficient, which is $15$.