Average total number of collisions

• Another way to look at the situation is this...
• You are throwing N balls randomly into M containers
• The first ball lands in some container. The remaining N-1 balls each have probability
  1/M of landing in the same container; so the average number of collisions with the first ball will be (N-1)/M
• The second ball lands in some container. The reminaing N-2 balls each have probability 1/M of landing in the same container; so the average number of total with the second ball will be (N-2)/M
• Etc... So the average total number of collisions is

• And so the expected (average) number of collisions will be 1 when:
  which for large M, implies