Question

Assertion (A) : The product of $10$ consecutive natural numbers is divisible by $9!$.
Reason (R) : The product of $n$ consecutive positive integers is divisible by $(n+1)!$.

• 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

Answer: (C)

$\because (m+1)(m+2)(m+3)...(m+n),m\in$ whole number
$=\frac{(1\cdot \mathrm{2....}m)(m+1)(m+2)....(m+n)}{1\cdot 2\cdot 3\cdot ....m}$
$=\frac{(m+n)!}{m!}=\frac{n!(m+n)!}{m!n!}$
$=n!({}^{m+n}{C}_n)=n!({}^{m+n}{C}_m)$
Which is divisible by $n!$, it is also divisible by $(n-1)!$ but not divisible by $(n+1)!$
$\therefore$ Assertion is true but Reason (R) is false.