Question

If $p_n$ denote the product of all coefficients in the expansion of ${(1+x)}^n$ ,then $\frac{p_{n+1}}{p_n}$ equals

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

Answer 1

Answer: (B)

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

$\frac{n+1C_r}{{}^n{C}_r}$
$=\frac{(n+1)!\left(r\right)!(n-r)!}{(n+1-r)!.\left(n\right)!\left(r\right)!}$
$=\frac{n+1}{n+1-r}$
Hence
$\frac{P_{n+1}}{P_n}$
$=\frac{n+1}{n+1}\frac{n+1}{n}\frac{n+1}{n-1}\times ...\frac{n+1}{1}$
$=\frac{(n+1{)}^{n+1}}{(n+1)!}$
$=\frac{(n+1{)}^n}{n!}$