Question

Assertion :$\frac{(n+3)!}{(n-1)!}$ is divisible by $24\left(nϵN\right)$ Reason: Product of any four consecutive integers is divisible by 4!

• 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),

$\frac{(n+3)!}{(n-1)!}$
$=n(n+1)(n+2)(n+3)$ ...(i)
Let $n=1$, then
$\left(1\right)\left(2\right)\left(3\right)\left(4\right)$
$=24$
Hence it is divisible by $24$.
Now using principle of mathematical induction, we can easily prove that it is divisible by $24$.
However $n(n+1)(n+2)(n+3)$ is nothing but a product of $4$ consecutive positive integers, as $nϵN$
Hence both assertion and reason is correct and reason is the correct explanation of assertion.