Question

Assertion :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 $\frac{{(n+1)}^n}{n!}$ Reason: ${}^{n+1}{C}_{r+1}=\frac{n+1}{r+1}{.}^n{C}_r$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given: $P_n={{}^{n}C}_0.{{}^{n}C}_1.{{}^{n}C}_2..{.}^n{C}_n$

Now, $\frac{P_{n+1}}{P_n}=\frac{{}^{n+1}{C_0.}^{n+1}{C}_1{.}^{n+1}{C}_2..{.}^{n-1}{C}_n}{{}^n{C_0.}^n{C_1.}^n{C_2...}^n{C}_n}$

$\Rightarrow \frac{P_{n+1}}{P_n}{=}^{n+1}{C}_0\left(\frac{{}^{n+1}{C}_1}{{}^n{C}_0}\right)\left(\frac{{}^{n+1}{C}_2}{{}^n{C}_1}\right)...\left(\frac{{}^{n+1}{C}_n}{{}^n{C}_n}\right){.}^{n+1}{C}_{n+1}$

Since, ${}^{n+1}{C}_{r+1}=\frac{n+1}{r+1}{.}^n{C}_r$

$\Rightarrow \frac{P_{n+1}}{P_n}=1\left(\frac{n+1}{1}\right)\left(\frac{n+2}{2}\right)...\left(\frac{n+1}{n}\right)1$

$\therefore \frac{P_{n+1}}{P_n}=\frac{{(n+1)}^n}{n!}$