<s>[INST] For what value of $m$ the is sum $\sum_{i = 0}^{m} {10 \choose i}{20 \choose m - i}$ where ${p\choose q} = 0$, if $pFor what value of $m$ the is sum
$$\sum_{i = 0}^{m} {10 \choose i}{20 \choose m - i}\text{where ${p\choose q}$} = 0\text{, if $p<q$, a maximum}$$
My approach
$$\sum_i^{m} {10 \choose i}{20 \choose m - i} = {10 \choose 0}{20 \choose m} + {10 \choose 1}{20 \choose m - 1} + \dots + {10 \choose m}{20 \choose 0}$$
$$(1 +x)^{20} = {20 \choose 0} + {20 \choose 1}x + \dots + {20 \choose m-1}x^{m-1} + {20 \choose m}x^{m} + \dots + {20 \choose 20}x^{20}$$
$$(1 +x)^{10} = {10 \choose 0} + {10 \choose 1}x + \dots + {10 \choose 10}x^{10}$$
Later what to do??
Any other method or hint will be greatly welcomed.
 [/INST] Write the (3) expansion as $$(1+1/x)^{10}=~^{10}C_0 + ~^{10}C_1 (1/x)+ ~^{10}C_2(1/x^2)+......+ ~^{10}C_n(1/x^{10})~~~~~(3)$$
Now multiply (2) and (3) mainly collecting the  terms independent of $x$ in the product $$S=\text{coefficient of $x^0$ in} (1+x)^{20}(1+1/x)^{10}$$
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