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Question
Find the middle terms(s) in the expansion of:
(iii) \[\left( 1 + 3x + 3 x^2 + x^3 \right)^{2n}\]
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Solution
\[(1 + 3x + 3 x^2 + x^3 )^{2n} \]
\[ = (1 + x )^{6n} \]
\[\text{ Here, n is an even number } . \]
\[ \therefore\text{ Middle term } = \left( \frac{6n}{2} + 1 \right)th = (3n + 1)\text{ th term} \]
\[\text{ Now, we have } \]
\[ T_{3n + 1} \]
\[ = ^{6n}{}{C}_{3n} x^{3n} \]
\[ = \frac{(6n)!}{(3n! )^2} x^{3n}\]
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