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Q.
In the expansion of $ {{(3x+2)}^{4}} $ the coefficient of middle term is:
Bihar CECEBihar CECE 2005
Solution:
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 $
$ = 3$ rd 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 $