<s>[INST] This sigma to binom? Can you please show me how to get from the left side to the right side? 
$$\sum\limits_{k=0}^{20}\binom{50}{k}\binom{50}{20-k} = \binom{100}{20}$$ 
 [/INST] $$(x+1)^{50}(1+x)^{50}=(1+x)^{100}$$
$$(\sum_{k=0}^{50}\binom{50}k x^k)\cdot(\sum_{k=0}^{50}\binom{50}kx^{50-k})=\sum_{k=0}^{50}\binom{100}r x^r$$
Consider the coefficients of $x^{20}$
$$\binom{100}{20}$$
$$=\binom{50}0\cdot\binom{50}{20}+\binom{50}1\cdot\binom{50}{19}+\cdots+
\binom{50}{19}\cdot\binom{50}1+\binom{50}{20}\cdot\binom{50}0$$
$$=\sum_{r=0}^{20}\binom{50}r\binom{50}{20-r}$$
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