$\begin{array}{1 1} 13 \\ 14 \\ 15 \\ 16 \end{array} $

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- $^mC_i$ is coefficient of $x^i$ in the expansion of $(1+x)^m$
- $^mC_r$ is maximum when $r=\large\frac{m}{2}$

$^{10}C_i$ is coefficient of $x^i$ in the expansion of $(1+x)^{10}$

and

$^{20}C_{m-i}$ is coefficient of $x^{m-i}$ in the expansion of $(1+x)^{20}$

$\therefore\: ^{10}C_i.\:^{20}C_{m-i}$ = Coefficient of $x^{i+m-i}$ in the expansion of $(1+x)^{10}.(1+x)^{20}$

$i.e.,$ Coefficient of $x^m$ in the expansion of $(1+x)^{30}$

$=^{30}C_m$

$\Rightarrow\:\displaystyle\sum _{i=o} ^{{m} ^{10}} C_i ^{20}C_{m-i}$ is maximum when $^{30}C_m$ is maximum

$\Rightarrow\:$ when $m=\large\frac{30}{2}$$=15$

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