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.