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Question: If (1 + x)<sup>n</sup> = \(\sum_{r = 0}^{n}C_{r}x^{r}\), then \(\left( 1 + \frac{C_{1}}{C_{0}} \righ...

If (1 + x)n = ∑r=0nCrxr\sum_{r = 0}^{n}C_{r}x^{r}, then (1+C1C0)(1+C2C1)........(1+CnCn–1)\left( 1 + \frac{C_{1}}{C_{0}} \right)\left( 1 + \frac{C_{2}}{C_{1}} \right)........\left( 1 + \frac{C_{n}}{C_{n–1}} \right)is equal to –

A

nn–1(n–1)!\frac{n^{n–1}}{(n–1)!}

B

(n+1)n–1(n–1)!\frac{(n + 1)^{n–1}}{(n–1)!}

C

(n+1)nn!\frac{(n + 1)^{n}}{n!}

D

(n+1)n+1n!\frac{(n + 1)^{n + 1}}{n!}

Answer

(n+1)nn!\frac{(n + 1)^{n}}{n!}

Explanation

Solution

(1+n1)(1+n(n–1)2n).............(1+1n)\left( 1 + \frac{n}{1} \right)\left( 1 + \frac{\frac{n(n–1)}{2}}{n} \right).............\left( 1 + \frac{1}{n} \right)

(1 + n) (n+12)(n+13)...........(n+1)n\left( \frac{n + 1}{2} \right)\left( \frac{n + 1}{3} \right)...........\frac{(n + 1)}{n}

= (n+1)nn!\frac{(n + 1)^{n}}{n!}