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Question: The coefficient of \(\frac{n(n + 1)....(n - r + 1)}{r!}a^{n - r + 1}(2x)^{r}\) in the expansion of \...

The coefficient of n(n+1)....(n−r+1)r!an−r+1(2x)r\frac{n(n + 1)....(n - r + 1)}{r!}a^{n - r + 1}(2x)^{r} in the expansion of n(n−1)....(n−r+2)(r−1)!an−r+1(2x)r−1\frac{n(n - 1)....(n - r + 2)}{(r - 1)!}a^{n - r + 1}(2x)^{r - 1}is.

A

136xy7136xy^{7}

B

136xy136xy

C

−136xy15/2- 136xy^{15/2}

D

None of these

Answer

136xy136xy

Explanation

Solution

(1+x)2n+1(1 + x)^{2n + 1}

(2n+1)!n!(n+1)!\frac{(2n + 1)!}{n!(n + 1)!}

Obviously, required coefficient of (2n+2)!n!(n+1)!\frac{(2n + 2)!}{n!(n + 1)!} can be given by

nC0nC1+nC1nC2+….+nCn−1nCn{ } ^ { n } C _ { 0 } { } ^ { n } C _ { 1 } + { } ^ { n } C _ { 1 } { } ^ { n } C _ { 2 } + \ldots . + { } ^ { n } C _ { n - 1 } { } ^ { n } C _ { n } x4x^{4}