If the coefficients of the r^{th} term and th | vedclass

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Question

(C) The general term in the expansion of $(1 + x)^n$ is given by $T_{k+1} = ^nC_k x^k$.For the $r^{th}$ term,$k = r - 1$,so the coefficient is $^{20}C

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(C) The general term in the expansion of $(1 + x)^n$ is given by $T_{k+1} = ^nC_k x^k$.For the $r^{th}$ term,$k = r - 1$,so the coefficient is $^{20}C_{r-1}$.For the $(r + 4)^{th}$ term,$k = r + 4 - 1 = r + 3$,so the coefficient is $^{20}C_{r+3}$.Given that the coefficients are equal,we have $^{20}C_{r-1} = ^{20}C_{r+3}$.Using the property $^nC_a = ^nC_b \implies a = b$ or $a + b = n$,we have:Case $1$: $r - 1 = r + 3$ (This leads to $-1 = 3$,which is impossible).Case $2$: $(r - 1) + (r + 3) = 20$.$2r + 2 = 20$.$2r = 18$.$r = 9$.

If the coefficients of the $r^{th}$ term and the $(r + 4)^{th}$ term are equal in the expansion of $(1 + x)^{20}$,then the value of $r$ is:

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