If the coefficients of the (2r + 1)^{th} term | vedclass

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Question

(A) The general term $T_{k+1}$ in the expansion of $(1 + x)^n$ is given by $\binom{n}{k} x^k$.The coefficient of the $(2r + 1)^{th}$ term is $\binom{4

Vedclass · Accepted Answer

$14$

Vedclass · Answer

(A) The general term $T_{k+1}$ in the expansion of $(1 + x)^n$ is given by $\binom{n}{k} x^k$.The coefficient of the $(2r + 1)^{th}$ term is $\binom{43}{2r}$.The coefficient of the $(r + 2)^{th}$ term is $\binom{43}{r+1}$.Given that these coefficients are equal,we have $\binom{43}{2r} = \binom{43}{r+1}$.Using the property $\binom{n}{a} = \binom{n}{b}$,we know that either $a = b$ or $a + b = n$.Case $1$: $2r = r + 1 \implies r = 1$.Case $2$: $2r + (r + 1) = 43 \implies 3r + 1 = 43 \implies 3r = 42 \implies r = 14$.Comparing with the given options,the value of $r$ is $14$.

If the coefficients of the $(2r + 1)^{th}$ term and the $(r + 2)^{th}$ term are equal in the expansion of $(1 + x)^{43}$,then the value of $r$ is:

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