Each of the digits 1,2,3,4,5,6 is used exactly once in forming the 3-digit integers X and Y. How many possible values of X + Y are there if X - Y = 333?
There's a way to simply brute force this, so it's not pretty, but it should be the right answer.
If each set of integers are three digits, and subtracting X from Y gives 333, we can assume that subtracting the single-digits in X (1/2/3) from Y (1/2/3) should yield 3. There's only three ways to get 3 using the numbers 1 through 6 with solely subtraction: 6-3, 5-2, and 4-1. We can therefore assume X (1/2/3) = 4, 5, and 6; and Y (1/2/3) = 1, 2, and 3.
It's a simple matter of lining up the ones, tens, and hundreds columns so that each column will spit out a 3. You have three columns, and three problem sets to apply to each column, which should result in nine different values for X/Y to go together to fulfill X - Y = 333.
I won't solve the whole thing for you, as I'm not sure whether the problem asks for fully simplified answers or just the problem sets, but I'll at least give you that little stepping stone.