ISI2015-MMA-9

Question

Let $(1+x)^n  = C_0+C_1x+C_2x^2+ \ldots +C_nx^n, \: n$ being a positive integer. The value of 

$$\left( 1+\frac{C_0}{C_1} \right) \left( 1+\frac{C_1}{C_2} \right) \cdots \left( 1+\frac{C_{n-1}}{C_n} \right)$$

is

• A. $\left( \frac{n+1}{n+2} \right) ^n$
• B. $ \frac{n^n}{n!} $
• C. $\left( \frac{n}{n+1} \right) ^n$
• D. $\frac{(n+1)^n}{n!}$

Answer 1

Here $C_k$ means $\binom{n}{k}$.

and $\binom{n}{k }$/$\binom{n}{k+1 }$=(k+1)/(n-k)

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Therefore,$C_0$/$C_1$=1/(n-0) and $C_1$/$C_2$=2/(n-1), $C_2$/$C_3$=3/(n-2)...

so (1+$C_0$/$C_1$)*(1+$C_1$/$C_2$)*(1+$C_2$/$C_3$)....(1+$C_n$/$C_n-1$) =(1+1/n)*(1+2/(n-1))*(1+3/(n-2)).....(1+n/(n-n+1))

=((n+1)/n)*((n+1)/(n-1))*((n+1)/(n-2))....*((n+1)/1)

=$(n+1)^{n}$/n!.

There fore option d is the right answer

Can some one edit this... i am new to latex