First let's look at what all the 60 combinations look like. Consider the first digit - obviously, we have 5 choices for which letter to place there. Since there are a total of 60 combinations, each letter must appear there a total of 12 times. Therefore, we can add up everything contributing to the sum from just that place alone as 1200*(A+B+C+D+E). This argument can be applied to each digit separately, so we can evaluate the entire sum as 1332*(A+B+C+D+E) since 12+120+1200=1332.
Now, the constraint that each letter must represent a distinct number means that the sum A+B+C+D+E must fall somewhere in the range of 15 (1+2+3+4+5) to 35 (9+8+7+6+5), inclusive. That's actually a small enough range to check by hand, in which case you get
as the unique answer for this problem.