How many collections of 12 Skittles have at least one of each color? (There are 5 colors of Skittles.)
Let the number of Skittles of each color be a, b, c, d, and e. We have 12 skittles in total:
a + b + c + d + e = 12
Note that we also need at least one of each color, so we can substitute a' = a + 1, b' = b + 1, and so on, to ensure that each color has at least one Skittle.
a' + b' + c' + d' + e' = 7
This is an example of stars and bars, and the ways to choose is (7+5-1) choose (7-1) = 462.