n(n+1)(n+2) will be divisible by 8 in the following 3 cases:

1: n is a multiple of 8

8, 16, 24, 32......96 - total 12 integers

2: n is an even number from 1-96 inclusive.

Total 48 integers. However since these 48 integers already include the above mentioned 12 multiples of 8 we won't include the previous 12 in the calculation in order to avoid redundancy.

3: n is an odd integer which is 1 less than the multiple of 8

7, 15, 31.....95 = 12 integers.

Total number of integers from 1-96 inclusive = 96

Hence total probability = (12+48 )/96 = 60/96 = 5/8.