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During his childhood, Utkarsh used to collect Pokemon Tazos. Pokémon Tazos were small collectible round discs that were found in bags of chips in the 1990s and 2000s. He wanted to collect all the different Pokemons Tazos.

He did some research and found out that there were a total of M distinct Pokemon Tazos. Each bag of chips contained exactly one tazo. One cannot see the tazo contained in the bag of chips before buying it. It was equally likely for any of the M distinct Pokemon Tazos to be present in a bag of chips.

In order to collect them all, he went to a shop and bought N bags of chips. Now he is worried whether he has bought enough bags of chips or not. You need to tell him the probability of getting all the M distinct Pokemon Tazos from the N bags of chips.

sandesh indavara
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1 Answers1

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What's the number of ways of arranging $N$ balls into $M<=N$ bins such that none of the $M$ bins is empty, or arranging sequentially $M-1$ bars and $N$ stars such that no bars are adjacent and no bar is at the end or beginning of the string? $N-1\choose M-1$, as any placement of $M-1$ bars in the $N-1$ gaps between stars gives rise to one of these arrangements. There are $M^N$ possible sets of cards that could come from the $N$ purchases. So the probability is ${N-1\choose M-1}/M^N.$

snarfblaat
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