An occupancy problem asks how objects, conventionally called balls, can be distributed among containers, conventionally called boxes (Feller 1968). Variants specify whether the balls and boxes are distinguishable or indistinguishable and whether empty boxes are allowed. These choices give the cases organized by the twelvefold way.
For 
distinguishable balls and 
distinguishable boxes, a placement has an occupancy vector
, where 
is the number of balls in box 
and 
. The number of placements having a specified occupancy
vector is the multinomial coefficient
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In probabilistic occupancy problems, a probability measure is specified on the set of possible assignments, and questions are asked about such quantities as the numbers of empty or multiply occupied boxes. The classical occupancy problem uses box choices that are independent and identically distributed according to the discrete uniform distribution. The coupon collector's problem asks how many independent ball placements are required before every box is occupied, while Dirichlet's box principle gives deterministic conditions forcing multiple occupancy.
