(2n+1)C0− (2n+1)C1+⋯+ 2n+1C2n=

2n+1C0− 2n+1C1+ 2n+1C2−⋯+ 2n+1C2n(1+x)2n+1= 2n+1C0+ 2n+1C1x+ 2n+1C2x2+⋯Put x=−10= 2n+1C0− 2n+1C1+ 2n+1C2− 2n+1C3+⋯+ 2n+1C2n− 2n+1C2n+1∴ 2n+1C0− 2n+1C1+ 2n+1C2− 2n+1C3+⋯+ 2n+1C2n=1

(2n+1)C0− (2n+1)C1+⋯+ 2n+1C2n=

2n+1C0− 2n+1C1+ 2n+1C2−⋯+ 2n+1C2n(1+x)2n+1= 2n+1C0+ 2n+1C1x+ 2n+1C2x2+⋯Put x=−10= 2n+1C0− 2n+1C1+ 2n+1C2− 2n+1C3+⋯+ 2n+1C2n− 2n+1C2n+1∴ 2n+1C0− 2n+1C1+ 2n+1C2− 2n+1C3+⋯+ 2n+1C2n=1

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$^{(2 n + 1)} C_{0} -^{(2 n + 1)} C_{1} + \dots +^{2 n + 1} C_{2 n} =$

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Detailed Solution

$\begin{array}{l} ^{2 n + 1} C_{0} -^{2 n + 1} C_{1} +^{2 n + 1} C_{2} - \dots +^{2 n + 1} C_{2 n} \\ {(1 + x)}^{2 n + 1} =^{2 n + 1} C_{0} +^{2 n + 1} C_{1} x +^{2 n + 1} C_{2} x^{2} + \dots \\ Put x = - 1 \\ 0 =^{2 n + 1} C_{0} -^{2 n + 1} C_{1} +^{2 n + 1} C_{2} -^{2 n + 1} C_{3} + \dots +^{2 n + 1} C_{2 n} -^{2 n + 1} C_{2 n + 1} \\ ∴^{2 n + 1} C_{0} -^{2 n + 1} C_{1} +^{2 n + 1} C_{2} -^{2 n + 1} C_{3} + \dots +^{2 n + 1} C_{2 n} = 1 \end{array}$