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Question Number 18759 by 786786AM last updated on 29/Jul/17

$$\mathrm{The}\mathrm{coefficient}\mathrm{of}\mathrm{the}\mathrm{middle}\mathrm{term}\mathrm{in}\mathrm{the}$$$$\mathrm{expansion}\mathrm{of}{(1+x)}^{2n}\mathrm{is}$$

Answered by anuragbhandari123@gmail.com last updated on 29/Jul/17

$$\frac{2nC(n+1)xofpower(n+1)}{}$$

Answered by Tinkutara last updated on 30/Jul/17

$$\mathrm{Number}\mathrm{of}\mathrm{terms}=2n+1$$$$\mathrm{Middle}\mathrm{term}={(n+1)}^{\mathrm{th}}\mathrm{term}$$$$T_{n+1}={}^{2n}{C}_n{x}^n$$$$\mathrm{Coefficient}\mathrm{of}\mathrm{middle}\mathrm{term}={}^{2n}{C}_n$$$$=\frac{\left(2n\right)!}{n!n!}=\frac{1.2.3.4....(2n-2)(2n-1)\left(2n\right)}{n!n!}$$$$=\frac{[1.3.5....(2n-1)][2.4.6....\left(2n\right)]}{n!n!}$$$$=\frac{[1.3.5....(2n-1)]2^n[1.2.3....\left(n\right)]}{n!n!}$$$$=\frac{1.3.5....(2n-1)}{n!}{2}^n$$

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