What if we wanted an upper bound restriction? For example, we might insist that no kid gets more than 4 cookies or that \(x, y, z \le 4\text{.}\) It turns out this is considerably harder, but still possible. The idea is to count all the distributions and then remove those that violate the condition. In other words, we must count the number of ways to distribute 11 cookies to 3 kids in which one or more of the kids gets more than 4 cookies. For any particular kid, this is not a problem; we do this using stars and bars. But how to combine the number of ways for kid A, or B or C? We must use the PIE.
in-context