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Question
If the sum of the binomial coefficients of the expansion \[\left( 2x + \frac{1}{x} \right)^n\] is equal to 256, then the term independent of x is
Options
1120
1020
512
none of these
Solution
1120
\[\text{ Suppose (r + 1)th tem in the given expansion is independent of x . } \]
\[\text{ Then, we have } \]
\[ T_{r + 1} = ^{n}{}{C}_r (2x )^{n - r} \left( \frac{1}{x} \right)^r \]
\[ = ^{n}{}{C}_r 2^{n - r} x^{n - 2r} \]
\[\text{ For this term to be independent of x, we must have } \]
\[n - 2r = 0\]
\[ \Rightarrow r = n/2\]
\[ \therefore \text{ Required term } = ^{n}{}{C}_{n/2} 2^{n - n/2} = \frac{n!}{\left[ \left( n/2 \right)! \right]^2} 2^{n/2} \]
\[\text{ We know } : \]
\[\text{ Sum of the given expansion } = 256\]
\[\text{ Thus, we have } \]
\[ 2^n . 1^n = 256\]
\[ \Rightarrow n = 8\]
\[ \therefore \text{ Required term } = \frac{8!}{\left( 4 \right)! \left( 4 \right)!} 2^4 = 1120\]
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