Guys, I ended up figuring it out :)
For those who want to know the answer:
It's n+2 choose 2. That's because you can let n be a number of pieces of candy, and let x, y, and z be the number of candies you give each of 3 kids. Now, applying the Hockey Stick Identity, the answer would be n+3-1 choose 3-1, which is n+2 choose 2.