Could someone please help me with this proof? Can't think of an idea

Question

Prove: $$\sum^k_{r=0}C^{r}_mC^{k-r}_n = C^{k}_{m+n}$$

Answer 1

Given: $$(1+x)^{m} = \sum_{r = 0}^{m} C^m_r x^m$$ $$(1+x)^{m}(1+x)^{n}=\sum_{r = 0}^{m} \sum_{h = 0}^{n} C^r_m C^{h}_n x^{r+h}=\sum_{i = 0}^{m+n} C^i_{m+n} x^{i} = (1+x)^{m+n}$$ Comparing coefficients of x by letting $i = r+h$ $$\sum_{r = 0}^{i} C^r_m C^{i-r}_n =C^i_{m+n} $$