Can you explain the relevance? If I put N items randomly into >> N buckets the chance of there being a second item in a particular bucket is small (as opposed to there merely being a bucket with two items, as in the birthday "paradox").

That doesn't apply here, since the birthday paradox is about the existence of a collision, not that any particular sequence collides.

Most people in the room will still have unique birthdays even if one pair share theirs.