The sum sumlimits_{i = 0}^m {binom{10}{i}} {b | vedclass

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(B) The given sum is $\sum\limits_{i = 0}^m {\binom{10}{i}} {\binom{20}{m - i}}$.By Vandermonde's Identity,this sum is equal to the coefficient of $x^

Vedclass · Accepted Answer

$15$

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(B) The given sum is $\sum\limits_{i = 0}^m {\binom{10}{i}} {\binom{20}{m - i}}$.By Vandermonde's Identity,this sum is equal to the coefficient of $x^m$ in the expansion of $(1+x)^{10} (1+x)^{20} = (1+x)^{30}$.Thus,the sum is equal to $\binom{30}{m}$.We know that $\binom{n}{m}$ is maximum when $m = \frac{n}{2}$ if $n$ is even,or $m = \frac{n-1}{2}$ and $m = \frac{n+1}{2}$ if $n$ is odd.Here,$n = 30$,which is even.Therefore,the sum $\binom{30}{m}$ is maximum when $m = \frac{30}{2} = 15$.Hence,the correct option is $B$.

The sum $\sum\limits_{i = 0}^m {\binom{10}{i}} {\binom{20}{m - i}}$,(where $\binom{p}{q} = 0$ if $p < q$),is maximum when $m$ is

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