Calculate the probability of getting a total of 6 in three throws of a die

Question

I am working on the following problem:

In three throws of a die, what is the probability of a total score of $6$?

My solution:
We can get $6$ by the combination $(4,1,1)$ which has $3$ permutations and $(3,2,1)$ which has $6$ permutations.
Therefore the probability is $$\frac{3 + 6}{6^3} = \frac{9}{216} = \frac{1}{24}$$ But the solution that is mentioned as correct in my notes is: $$\frac{5}{108}$$

What am I doing wrong here?

Answer 1

Your approach is correct, but you forgot the combination $(2,2,2)$.