Factors

There will be 96 different possible numbers corresponding to the 96 different "n" that could be picked. All the n have equal likelihood.

look at first term in n(n+1)(n+2)

plugging values into n(n+1)(n+2)

If n=1 then 1*2*3 -> not divisible
n=2 -> 2*3*4 -> divisible
n=3 -> 3*4*5 -> not div
n=4 -> 4*5*6 -> div

form this all even n will result in divisibilty by 8....96/2=48 even numbers so 48 numbers

but it is also possible that n=7 will resuly in div by 8

n=7 -> 7*8*9...

if you keep adding 8 to n=7 all those numbers are div by 8. there are 12 og those numbers so a further 12 numbers

do the same with n+2 and n=6.......you will find all numbers are div by 8...but these n are all even so covered in the first 48

so probability is (48+12)/96=5/8

If n is an integer from 1 to 96 (inclusive), what is the probability that n*(n+1)*(n+2) is divisible by 8?

A.1/4
B.1/2
C.5/8
D.7/8
E.3/4

Case 1: n(n+1)(n+2) = even*odd*even = multiple of 8:
Since every other even integer is a multiple of 4, the product here will always include an even integer and a multiple of 4, resulting in a multiple of 8.
Thus, n can be any even integer between 1 and 96.
96/2 = 48 favorable choices for n.

Case 2: n+1 is a multiple of 8:
The product will be a multiple of 8 if n+1 is a multiple of 8.
Number of multiples of 8 between 1 and 96 = 96/8 = 12.
Thus, there are 12 favorable choices for n+1, implying that there are 12 more favorable choices for n.

Total favorable choices for n = 48+12 = 60.
Favorable choices/Total choices = 60/96 = 5/8.

The correct answer is C.

Alternate solution:

n(n+1)(n+2) = the product of 3 consecutive integers.
WRITE IT OUT and LOOK FOR A PATTERN.

1*2*3
2*3*4
3*4*5
4*5*6
5*6*7
6*7*8
7*8*9
8*9*10

9*10*11
10*11*12
11*12*13
12*13*14
13*14*15
14*15*16
15*16*17
16*17*18

Each of the products in red is a multiple of 8.
The two examples above imply the following:
Of every 8 products, exactly 5 will be a multiple of 8.

Thus, the probability that n(n+1)(n+2) will be a multiple of 8 = 5/8.

For a similar problem, check here:

https://www.beatthegmat.com/probability-t116280.html

Second approach is fabulous; probability is answered soon after reading the pattern. Further, listing the situations down is the safest and surest approach on those probability questions on GMAT which might be mathematically too nasty.

n(n+1)(n+2) is the product of 3 consecutive integers.
Between 1 and 8 there are exactly 5 values of n that return a multiple of 8:
2*3*4, 4*5*6 , 6*7*8, 7*8*9 , 8*9*10
Therefore 5/8 of the full sample (= 12 batches of 1 to 8) will comply.
Hence P = 5/8.