How many ways to divide 7 students into 3 groups with 2 groups containing 2 people each and 1 group containing 3 people?

Question

How many ways can you divide 7 students into three teams where two of the teams contain two people and one of the teams contains three?

I got $\frac{7!}{2!2!3!}$ but the correct answer was $\frac{7!}{2!3!2!2!}$.

Why is there an extra 2!?

Answer 1

You thought right just a bit incomplete. The answer you got shows that that you divide seven students in

(7!)/(2!2!3!).

Suppose the 2 groups which have 2 people each is labeled as G1 and G2.

Now the labeling could have been G2 and G1 also.

There are 2! ways in which you can label the groups having 2 people each !

If there had been say 3 groups of a certain number of students where each group had 3 people. Then in the denominator you would have got

3!3!3!3! .

The extra 3! comes from the 6 possible arrangements of the 3 groups themselves.

Like say you have ABCDEF and you have to distribute this into 3 groups of 2 each.

One such group be

AB CD EF

Now this group can be also written as

CD EF AB , EF AB CD , CD AB EF , AB EF CD , EF CD AB. ( 3! =6 Ways )

Similarly for each possible combination of groups.

That's why you get an extra 2! ( or 3! ) term due to arrangement of the groups.

Answer 2

We have to treat the two partitions say $(AB)(CD)(EFG)$ and $(CD)(AB)(EFG)$ as equal.