Assertion (A ): ((n + 3)!/(n - 1)!) is divisible by 24 (n ∈ N). Reason (R ): Product of any four consecutive integers is divisible by 4!.

Question

Assertion (A ): ((n + 3)!/(n - 1)!) is divisible by 24 (n ∈ N). Reason (R ): 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.

Tardigrade Admin · Accepted Answer

((n + 3)!/(n - 1)!) = ((n + 3)(n + 2)(n + 1)n(n-1)!/(n -1)!) = n (n+ 1) (n + 2) (n + 3) is the product of four consecutive integers it is divisible by 24.

Q. Assertion (A ): $\frac{( n + 3 )!}{( n - 1 )!}$ is divisible by $24 (n \in N)$ .
Reason (R ): Product of any four consecutive integers is divisible by $4!$ .

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

Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

(A) is true but (R) is false.

(A) is false but (R) is true.

$\frac{( n + 3 )!}{( n - 1 )!} = \frac{( n + 3 ) ( n + 2 ) ( n + 1 ) n ( n - 1 )!}{( n - 1 )!}$
$= n (n + 1) (n + 2) (n + 3)$ is the product of four consecutive integers & it is divisible by $24$ .

Q. Assertion (A ): (n−1)!(n+3)!​ is divisible by 24(n∈N). Reason (R ): Product of any four consecutive integers is divisible by 4!.