Tom has 15 ping-pong balls each uniquely numbered from 1 to 15 and also has a red box, a blue box and a green box. In how many ways Tom place the 15 distinct balls into the three boxes so that no box is empty.

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Detailed Solution

There are 15 ping pong balls, each with a unique number from 1 to 15. Additionally, we have three red, blue and green boxes. So not all balls and not all boxes are the same.
For section A, we need to think of several ways that Tom could place three different boxes with 15 different balls, making sure that no box is left empty. Let's start by placing one ball in each of the three boxes. Tom can choose 3 different balls from a pool of 15 balls in different ways, for example, 15C3. (nCr gives the total number of possible ways r items can be selected from n items.)
Tom puts these balls in each box to ensure that each box is not empty. He can do it in 3! ways then. Now he can place these 12 marbles in any way he likes on the 3 boxes. The total number of ways he can give these 12 balls becomes