Let be the number of balls in box number , where .
Then has a Binomial distribution with , so
otherwise, for ,
Then the expected number of boxes with no more than one ball in each box is
where is given by (*) above.
The theorem that E(X+Y) = E(X) + E(Y) plays a crucial role at equation (**).
It's important to know that this theorem is true even if X and Y are not independent, which is good for us here because the 's are not independent.