As $n(n^{2}-1)$ = (n−1) n (n+1)

Since n is any odd number , n−1 and n+1 are consecutive even integers.

So, one is divisible by 2 and the other is divisible by 4.

Hence, (n−1) n (n+1) is divisible by 8.

Since n−1, n , n+1 are 3 consecutive integers, one of them will be divisible by 3.

Hence, (n−1) n (n+1) is divisible by 3.

Since (n−1) n (n+1) is divisible by both 8 and 3, it is divisible by lcm(8,3)=24.

**then the largest number that divides $n(n^{2}-1)$ is 24**

**Hence Option C is the Correct Answer.**