Question

Assertion :(A): If $A=(300{)}^{600},B=600!,C=(200{)}^{600}$, then $A>B>C$. Reason: (R): ${\left(\frac{n}{2}\right)}^n>n!>{\left(\frac{n}{3}\right)}^n$ for $n>6$.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).

$\frac{n}{2}>\frac{n}{3}\Rightarrow {\left(\frac{n}{2}\right)}^n>{\left(\frac{n}{3}\right)}^n$

Also, $\frac{n}{2}\times \frac{n}{2}\times$...$\times \frac{n}{2}\times \frac{n}{2}>n\times (n-1)\times (n-2)\times$...$\times 2\times 1$

$\Rightarrow {\left(\frac{n}{2}\right)}^n>n!$

Hence, option $A$ is correct.