Vandermonde's identity

Algebraic (coefficient extraction). The product expands as (1+x)^m (1+x)^n = (1+x)^{m+n}. The coefficient of x^r on the left is Σₖ C(m,k) C(n, r−k) (choose k from the first factor and r−k from the second), while on the right it is C(m+n, r). Hence the two are equal.

Combinatorial. To choose r elements from a disjoint union of an m-set and an n-set, first decide that exactly k come from the first set (there are C(m,k) ways) and the remaining r−k from the second (there are C(n,r−k) ways). Summing over all admissible k counts each r-subset exactly once, giving C(m+n,r).

Bounds of summation. Nonzero terms occur only when 0≤k≤m and 0≤r−k≤n, that is max(0, r−n) ≤ k ≤ min(r, m).