Question

Let $P_n={\prod }_{k=0}^n{C}_k$ that is, $P_n$ denote the product of all the binomial coefficients in ${(1+x)}^n$, then the ratio $\frac{P_{n+1}}{P_n}$ equals

• A. $\frac{n^n}{n!}$
• B. $\frac{{(n+1)}^n}{n!}$
• C. $\frac{{\left(n\right)}^n}{(n+1)!}$
• D. none of these

Answer 1

The correct option is B $\frac{{(n+1)}^n}{n!}$

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

Hence,
$\frac{P_{n+1}}{P_n}$ $=\frac{n+1}{n+1}\frac{n+1}{n}\frac{n+1}{n-1}\times \cdots \frac{n+1}{1}$
$=\frac{(n+1{)}^{n+1}}{(n+1)!}$ $=\frac{(n+1{)}^n}{n!}$