Question

Assertion :The number ${}^{1000}{C}_{500}$ is not divisible by $11$, because Reason: If $p$ is a prime, the exponent of $p$ in $n!$ is $\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\left[\frac{n}{p^3}\right]+...$ where $\left[x\right]$ denotes the greatest integer $\le x$

• 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

For the truth of statement $-2$, see theory, page $\mathrm{5..3.}$

We have

$\left({}^{1000}{C}_{500}\right)=\frac{1000!}{500!500!}$

Exponent of $11$ in $1000!$ is $\left[\frac{1000}{11}\right]+\left[\frac{1000}{{11}^2}\right]=90+8=98$

Exponent of $11$ in $500!$ is

$\left[\frac{500}{11}\right]+\left[\frac{500}{{11}^2}\right]=45+4=49$.

Thus, exponent of $11$ in $\left({}^{1000}{C}_{500}\right)$ is $0$.

$\Rightarrow \left({}^{1000}{C}_{500}\right)$ is not divisible by $11$.