Question

Assertion :The product of 10 consecutive natural numbers is divisible by 9!. Reason: 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

The correct option is C (A)is true but (R) is false,

$(m+1)(m+2)(m+3)....(m+n)=(mϵ$ whole number)

$\frac{\left(\mathrm{1.2.....}m\right)(m+1)(m+2).....(m+n)}{1\cdot 2\cdot \mathrm{3......}m}$ $\frac{(m+n)!}{m!}=\frac{n!(m+n)!}{m!n!}=n!{(}^{m+n}{C}_n)=n!{(}^{m+n}{C}_m)$
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.