There are N cards on the table, whose front side is writen one integer number from 1 to M. We call one card "a type k card" if its number is k. The quantity of type i cards is a_i.
Let's play a game with these cards. We divide these cards into M piles by random with the only constrains that the quantity of cards in i-th (indexed from 1) pile must exactly be a_i. The possbility of each card appears in i-th pile is directly proportional to the size of this pile. That is to say, if the size of a pile is A, the possibility for each card appears in this pile is A/N assuming that N is the amount of all cards. We choose pile 1 to start the game. Assuming the we now play this game at pile k, we randomly choose a card from pile k with the same possibility for all cards in it, remember the number written on this card and throw it away. If the number on the chosen card is j, we continue this game at pile j on next round. The game terminates when we are going to get a card from an empty pile.
Now the question is, when the game ends, what is the possibility that all piles are empty?