If there is a quicker way to do this please let me know! Took me a little longer than I would have liked

Key Point #1: n(n+1)(n+2) = multiplying consecutive integers
Key Point #2: When multiplying consecutive integers, the product will always be divisible by 8 when there is an odd number in the middle:

Example: 2x3x4 = 24 (24/8=3)
Example: 4x5x6 = 120 (120/8 = 15)

Step 1:
We know that all the groups of 3 consecutive numbers that start with an even number will be divisible by 8. Since the range is 1-72, we can just divide 72 by 2 to get total # of even numbers (these will start a consecutive group of 2 and be divisible by 8).

72/2 = 36

Step 2
There are a groups of consecutive integers that are odd and are divisible by 8. These are groups where the middle number (n+1) is divisible by 8.

Example: 7x8x9 = 504 (504/8 = 63)

There are 9 consecutive integer groups that a) start with an odd number and b) be divisible by 8.

Middle numbers (n+1): 8, 16, 24, 32, 40, 48, 56, 64, 70 (alternatively, you can divide 72/8)

There are 36 groups of 3 consecutive integers that start with an even integer whose product is divisible by 8
There are 9 groups of 3 consecutive integers that start with an odd integer whose product is divisible by 8
36+9 = 45 total groups of 3 consecutive integers whose product is divisible by 8


Step 3
The questions asks for the probability that n(n+1)(n+2) will be divisible by 8.

Total number of eligible groups/total number of possible consecutive groups of 3 = 45/72 = 5/8 (after diving the numerator and denominator by 9)