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 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.