Counting the number of combinations from 5 equals sets

Question

I have 5 sets with three elements each:

A can be either 2,1 or 0
B can be either 2,1 or 0
.
.
E can be either 2,1 or 0

You can only pick one element from one of the sets.

For instance, I know there is only one way of obtaining 10 and that is by picking the element 2 from the 5 sets.

But I want to know how many combinations there is for example to get 5? And what would be the probability for this? I would need to get the total number of possible combinations, but I don't know how to obtain this?

I thought of 15 choose 5. But that is way too many combinations as I can only pick one element from each set.

Answer 1

I believe I have an answer. Let's start by considering the number of possible states. There should be $3^5 = 243$ , one state is one choice of element for each set. To get $5$ we would look at how to get there: {2+2+1},{2+1+1+1},{1+1+1+1+1}

How many states are there for these? ${5}\choose{2,1,2}$ + ${5} \choose {1,3,1}$ + ${5} \choose {0,5,0}$ $= 30 +20+1=51$

The probability is thus $51/243 \approx 0.21$

This method seems laborious, so if someone has something better I would appreciate it.