(C) The given expression is $S = \sum_{i=0}^{m} \binom{10}{i} \binom{20}{m-i}$.By Vandermonde's Identity,this sum represents the coefficient of $x^m$

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(C) The given expression is $S = \sum_{i=0}^{m} \binom{10}{i} \binom{20}{m-i}$.By Vandermonde's Identity,this sum represents the coefficient of $x^m$ in the expansion of $(1+x)^{10} (1+x)^{20} = (1+x)^{30}$.Thus,$S = \binom{30}{m}$.The binomial coefficient $\binom{n}{r}$ is maximum when $r = \frac{n}{2}$ if $n$ is even,or $r = \frac{n \pm 1}{2}$ if $n$ is odd.Here,$n = 30$,which is even.Therefore,the maximum value occurs when $m = \frac{30}{2} = 15$.

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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If $\binom{p}{q} = {}^{p}C_{q}$ and $\sum_{i=0}^{m} \binom{10}{i} \binom{20}{m-i}$ is maximum,then $m=$

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$10$
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$12$
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$15$
D
$20$

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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