Method 1: Observe that there are two ways to write $5$ as a sum of three positive integers.
5 & = 3 + 1 + 1\\
& = 2 + 2 + 1
$3 + 1 + 1$: Choose which of the three letters appears in the string three times. Choose three of the five positions for that letter in the string. Arrange the remaining two letters in the remaining two positions.
$2 + 2 + 1$: Choose which letter appears once. Choose in which of the five positions that letter appears. That leaves four positions to be filled with the two remaining letters. Choose which two of the remaining four positions will be occupied by the remaining letter that appears first in an alphabetical list. The remaining two positions must be filled with the remaining letter.
Method 2: We use the Inclusion-Exclusion Principle.
If there were no restrictions, there would be three ways to fill each of the five positions. From these $3^5$ sequences, we must exclude those in which not all three letters are used.
There are three ways to exclude one of the letters. For each such choice, the remaining positions can be filled in two ways.
However, if we subtract these cases from the total, we will have subtracted each case in which only one letter is used twice, once for each of the two ways we could designate one of the other letters as the excluded letter. Since we only want to subtract these sequences once, we must add these three sequences to the total.