How Many Ways Can The Class Be Divided Into Groups With Equal Numbers Of Students?

Question

How Many Ways Can The Class Be Divided Into Groups With Equal Numbers Of Students?

2 Answers

Answer Question

Answer 1

What is the number of students and how many factor pairs does that number have? for example...if you have 24 students, the factor pairs are 1 and 24, 2 and 12, 3 and 8, and 4 and 6.

With this size class, you can have...

1 group of 24
24 groups of 1
2 groups of 12
12 groups of 2
3 groups of 8
8 groups of 3
4 groups of 6 OR
6 groups of 4

Answer 2

It depends on the number in the class and the number in the group.

If your class has 22 students, and there are 11 in each group, there are 352,716 ways to choose sides. If your class has 21 students, and there are 7 in a group, there are 66,512,160 ways to choose teams. (I may have lost count around 50 million somewhere. LOL)

I believe the number you seek is the product of
- the number of combinations of (remaining class size) taken (group size) at a time, where (remaining class size) is the size of the class before the current group is chosen.

That product then needs to be divided by the number of permutations of (class size)/(group size) things taken that many at a time.

If we let C = the number of students in the class, and G = the number of students in the group, the number of ways of forming teams (T) according to the above calculation method reduces to
T(G,C) = C!/(((G!)^(C/G))*(C/G)!)

Some "representative" numbers:
T(10,5) = 126
T(10,2) = 945
T(22,11) = 352,716
T(21,7) = 66,512,160
T(21,3) = 36,212,176,000