Question

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

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

Hint:

Use properties of binomial coefficients and factorial simplifications for product formulas.

Answer 1

The Correct Option is D

The binomial coefficients for $(1 + x)^n$ are: \[ \binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}. \] The product $P_n = \prod_{k=0}^n \binom{n}{k}$. Using known formula or induction: \[ \frac{P_{n+1}}{P_n} = \frac{(n+1)^{n+1}}{(n+1)!}. \] Hence, option (4) is correct.