15 identical objects are distributed at random into 4 boxes, numbered 1,2,3 & 4. Find the probability that 1. Each box contains at least 2 objects
2. No box is empty

The number of ways to distribute r identical objects into n distinct boxes is given by

C(r+n-1,r). In your case, C(18,15)=816 total ways.


To find the number of ways to have at least two in each, put two balls in each of the boxes and then distribute the other 7 among the 4 boxes.

Also, do you know about generating functions? The most we can have in one box and still have at least two in the other 3 is 9.

So, we can use the generating function \left(\sum_{k=2}^{9}x^{k}\right)^{4}

Expand, then look at the coefficient of the 15th power of x.

For the second part, think about it. In order for no box to be empty, there must be at least one in each box.