If $P_{n}$ denotes the product of the binomial coefficients in the expansion of $(1 + x)^{n},$ then $\frac{P_{n + 1}}{P_{n}}$ equals

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$\frac{n + 1}{n!}$

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

$\frac{n^{n}}{n!}$

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

answer is D.

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Detailed Solution

We have,

$P_{n} =^{n} C_{0} \cdot^{n} C_{1} \cdot^{n} C_{2} \dots^{n} C_{n}$

and, $P_{n + 1} =^{n + 1} C_{0} \cdot^{n + 1} C_{1} \cdot^{n + 1} C_{2} \dots^{n + 1} C_{n + 1}$

$\begin{array}{l} ∴ \frac{P_{n + 1}}{P_{n}} = \frac{^{n + 1} C_{0} \cdot^{n + 1} C_{1} \cdot^{n + 1} C_{2} \dots^{n + 1} C_{n + 1}}{^{n} C_{0} \cdot^{n} C_{1} \cdot^{n} C_{2} \dots^{n} C_{n}} \\ \Rightarrow \frac{P_{n + 1}}{P_{n}} = (\frac{^{n + 1} C_{1}}{^{n} C_{0}}) \cdot (\frac{^{n + 1} C_{2}}{^{n} C_{1}}) \cdot (\frac{^{n + 1} C_{3}}{^{n} C_{2}}) \dots (\frac{^{n + 1} C_{n + 1}}{^{n} C_{n}}) \end{array}$

$\Rightarrow \frac{P_{n + 1}}{P_{n}} = (\frac{{\frac{n + 1}{n}}^{n} C_{0}}{^{n} C_{0}}) (\frac{{\frac{n + 1}{2}}^{n} C_{1}}{^{n} C_{1}}) (\frac{\frac{n + 1}{3} \cdot^{n} C_{2}}{^{n} C_{2}}) \dots (\frac{n + 1 n^{n} C_{n}}{^{n} C_{n}})$

$\Rightarrow \frac{P_{n + 1}}{P_{n}} = (\frac{n + 1}{1}) (\frac{n + 1}{2}) (\frac{n + 1}{3}) \dots (\frac{n + 1}{n + 1}) = \frac{(n + 1)^{n + 1}}{(n + 1)!}$