Formula for counting ways to divide a number of people into separate groups Assume six people at a party.  Is there a formula to calculate the total possible combinations?  Ie:  Six alone.   Four together, two alone.  Four together, two together. 3 together, 3 others together. 3 together, 2 together, 1 alone and so on?  I came up with 10 distributions and 142 combinations for that (which might be wrong).  
I'm more interested in expanding that to other quantities.  
 A: Your "distributions" are commonly called "partitions". These are an extremely well-studied object in research-level mathematics.
The short answer is no, there is no such formula.
The longer answer is that there actually are such formulae, but they are all somewhat unsatisfying.


*

*There are recursive rules which give you partition numbers if you know all of the previous partition numbers. 

*There are also approximate formulae which can be off by very large amounts but the error is guaranteed to be small compared to the size of the numbers involved (think Earth vs. the Milky Way. Yes, Earth is big, but it's pretty small on a galactic scale).

*There are formulae which involve adding infinitely many numbers together, with the partial sums eventually converging on a single number.

*In 2011, Jan Bruinier and Ken Ono found an exact, closed form, finite formula, but if you want to actually use it to compute partition numbers then you will need to be willing to learn some very, very deep mathematics.


I think your "combinations" are ways of first partitioning ("distributing") the people, and then determining which people go in which groups? 
If you are allowed to distinguish the groups themselves (e.g. ABC/DEF is different than DEF/ABC), then this is simply the number of surjections from $\{1,\dots, n\}$ to $\{1,\dots, k\}$, summed over all $k$; I think these are easily countable by standard methods.
However, my guess is that you would want to make the groups indistinguishable. This may also have a simple answer, but I am not aware of it.
