A) 36
B) 216
C) 54
D) 81
Correct Answer: B
Solution :
Key Idea: In the expansion of \[{{(1+x)}^{n}},\] the middle term is\[\frac{n}{2}th\] if \[n\] is even and \[\left( \frac{n}{2}+1 \right)th\]if \[n\]is odd. In the given expansion \[{{(3x+2)}^{4}}\] Here, \[n=4\] \[\therefore \] Middle term \[=\frac{n}{2}+1,=\frac{4}{2}+1\] \[=\text{ }3\text{ }rd\text{ }term\] \[\therefore \] \[{{T}_{3}}={{T}_{2+1}}\] \[={{\,}^{4}}{{C}_{2}}{{(3x)}^{2}}{{(2)}^{2}}\] The coefficient of middle term\[={{\,}^{4}}{{C}_{2}}{{3}^{2}}{{2}^{2}}\] \[=6\times 9\times 4=216\]
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