Thread: partitioning in a number of groups

1. partitioning in a number of groups

Someone know how to compute in how many ways is it possible to partition a collection of $\displaystyle M$ indistinguishable elements in $\displaystyle I$ groups (evidently with at least one element for each groups)???
The groups are distinguishable; I mean, if $\displaystyle M=3$ and $\displaystyle I=2$ the solution is 2 combinations (1,2) and (2,1) (whereas (3,0) and (0,3) are not allowed since a group is empty)

( probably related to this thread
http://www.mathhelpforum.com/math-he...subgroups.html )

2. $\displaystyle {{M-1}\choose{I-1}}$

3. Thank you very much...
and admitting also an indefinite number of "empty" groups within the total number of groups $\displaystyle I$? so, in the same example of above, if $\displaystyle M=3$ and $\displaystyle I=2$ the solution is 4 combinations: (3,0), (2,1), (1,2), (0,3).

4. Allowing for groups to be empty: $\displaystyle {{M+I-1}\choose{M}}$ .

5. the last question...I hope
is there also a simple form for to give an answer to the previous two posts in the case of distinguishable elements?
So, the number of combinations of $\displaystyle M$ distinguishable elements in $\displaystyle I$ distinguishable groups, with and without admitting for empty groups.
For example, if $\displaystyle M=2$ and $\displaystyle I=2$:
- without admitting for empty groups: $\displaystyle (obj_1, obj_2), (obj_2, obj_1) == 2$ combinations.
- admitting empty groups: $\displaystyle (obj_1, obj_2), (obj_2, obj_1), (obj_1 \& obj_2, nothing), (nothing, obj_1 \& obj_2) == 4$ combinations.

Thanks!