N balls and M boxes, probability that there is at least 1 box contains at least 2 balls 
I have $N$ balls and $M$ boxes. Balls are thrown to the boxes at random.
  What is the probability that there is at least 1 box contains at least 2 balls?

Thank you very much

The complementary event $A$ is that all $M$ boxes contain $0$ or $1$ balls. There are $\binom{M}{N}$ possibilities for this (assuming this to be zero if $N>M$). Overall, we have $R=\binom{N+M-1}{M-1}$ possibilities to distribute $N$ balls on $M$ boxes, so $P(A)=\frac{\binom{M}{N}}{R}$ and thus the probability of your event is $1-P(A)$.
