Occupancy theorem

In combinatorial mathematics, the occupancy theorem states that the number of ways of putting r indistinguishable balls into n buckets is

<math> = .</math>

Furthermore, the number of ways of putting r indistinguishable balls into n buckets, leaving none empty is

<math> = .</math>

Applications

This has many applications in many areas where the problem can be reduced to the problem stated above.

For example:Take 12 red and 3 yellow cards, shuffle them and deal them in such a way that all the red cards before the first yellow card go to player 1, between the 1st and 2nd second yellow cards go to player 2, and so on.

Q: Find Pr(Everyone has at least 1 card)

A: The number of allocations of 12 balls (red cards) to 4 buckets (players) is <math>15 choose 3</math>. The number of allocations where each player gets at least one card is <math>11 choose 3</math>, so the probability is <math>frac = frac</math>.

See also

• Multiset coefficients for an explanation of the result

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