There are $7776$ possible outcomes of rolling $5$ six-sided dice, of this total there are $651$ possible outcomes where the $5$ dice equal exactly $15$.
How could you calculate the number of possible outcomes where the top three dice equal $15$?
There are three ways to roll 15 with three dice: 663, 654 and 555. I'll call this the greatest-sum-set (GSS); these have to be the greatest within the roll. I will split each case into three sub-cases, depending on how many times the smallest number in the GSS appears on the remaining two dice:
Adding up all cases gives 1111 5-dice rolls where the top three numbers sum to 15.