(B) The given expression is the sum of products of combinations: $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ... + ^{20}C_0 ^{20}C_r = ^{40}C_r$ This

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(B) The given expression is the sum of products of combinations: $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ... + ^{20}C_0 ^{20}C_r = ^{40}C_r$ This is based on Vandermonde's Identity,which states that $\sum_{k=0}^{r} {^nC_k} {^mC_{r-k}} = ^{n+m}C_r$. Here,$n=20$ and $m=20$,so the sum is $^{40}C_r$. The value of $^{40}C_r$ is maximum when $r = \frac{n+m}{2} = \frac{40}{2} = 20$.

Class 11 Mathematics Permutation and Combination Definition of combinations, Condition combinations

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The value of $r$ for which $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ^{20}C_{r-2} ^{20}C_2 + ... + ^{20}C_0 ^{20}C_r$ is maximum is

JEE MAIN 2019 All JEE MAIN

A
$15$
B
$20$
C
$11$
D
$10$

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Permutation and Combination

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Definition of combinations, Condition combinations

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The value of $r$ for which $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ^{20}C_{r-2} ^{20}C_2 + ... + ^{20}C_0 ^{20}C_r$ is maximum is

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