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 N1 balls each have probability
1/M of landing in the same container; so the average number of collisions with the first ball will be (N1)/M

The second ball lands in some container. The reminaing N2 balls each have probability 1/M of landing in the same container; so the average number of total with the second ball will be (N2)/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
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