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Question: <sup>n</sup>C<sub>0</sub> – \(\frac{nC_{1}}{2}\) + \(\frac{nC_{2}}{3}\) – \(\frac{nC_{3}}{4}\) + ......

nC0 – nC12\frac{nC_{1}}{2} + nC23\frac{nC_{2}}{3} – nC34\frac{nC_{3}}{4} + ......... =

A

0

B

2n–1

C

1n+1\frac{1}{n + 1}

D

None of these

Answer

1n+1\frac{1}{n + 1}

Explanation

Solution

∫–10(1+x)n\int_{–1}^{0}{(1 + x)^{n}}=∫–10nC0+nC1x+nC2x2+......+nCn 6muxn\int_{–1}^{0}{nC_{0} +^{n}C_{1}x +^{n}C_{2}x^{2} + ...... +^{n}C_{n}\mspace{6mu} x^{n}} ∣(1+x)n+1n+1∣–10\left| \frac{(1 + x)^{n + 1}}{n + 1} \right|_{–1}^{0} = nC0 – nC12\frac{nC_{1}}{2}+ nC23\frac{nC_{2}}{3} ..........

= 1n+1\frac{1}{n + 1}