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How many ways can I combine/group N items? Example of what I mean:

Items a,b,c: (abc) / (ab)(c) / (a)(bc) / (ac)(b) / (a)(b)(c) = 5 where parenthesis mean a group distinct in some way from the others.

Or for items a,b,c,d,e,f one might have: (ae)(b)(cdf)

In other words, all breakdowns, of variable group sizes.

(I did search, but either it hasn't been dealt with here or I do not know the correct mathematical terminology.)

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The mathematical term is "set partition." The number of ways to partition a set with $n$ elements is $B_n$, the $n$th Bell number (not to be confused with Bernoulli numbers). – anon Jun 26 '13 at 3:15
@anon Perhaps you should post that as an answer. – Alex Becker Jun 26 '13 at 3:18
What questions are you referring to? – anon Jun 26 '13 at 3:19
Ok. Followed your link to wikipedia. Just what I was looking for. Thanks. – Jeff Jun 26 '13 at 3:20

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