The value of r for which ^{20}C_r cdot ^{20}C | vedclass

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(B) The given expression is $^{20}C_r \cdot ^{20}C_0 + ^{20}C_{r-1} \cdot ^{20}C_1 + \dots + ^{20}C_0 \cdot ^{20}C_r$.By Vandermonde's Identity, this

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$20$

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(B) The given expression is $^{20}C_r \cdot ^{20}C_0 + ^{20}C_{r-1} \cdot ^{20}C_1 + \dots + ^{20}C_0 \cdot ^{20}C_r$.By Vandermonde's Identity, this sum is equivalent to the coefficient of $x^r$ in the expansion of $(1+x)^{20} \cdot (1+x)^{20} = (1+x)^{40}$.Thus, the sum is equal to $^{40}C_r$.The binomial coefficient $^{n}C_r$ is maximum when $r = n/2$.For $n = 40$, the maximum value occurs at $r = 40/2 = 20$.

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

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