I hope I can explain this clearly.
1. Twenty-five girls are bored. .How many ways may the girls
form two teams of 5 to play basketball against each other?
I used and got
The answer in the back says it is 411,863,760.
It might just be coincidence but the answer in the back is half the number I got.
Let's name the twenty-five girls: Ann, Beth, Carol, Doris, . . . Yvonne.
Also, let's name the two teams: Reds and Blues.
is the number of ways to select the Red Team.
Then is the number of ways to select the Blue Team.
The product is the number of ways to form the two teams.
Included in this long list are these two partitions:
Since the two teams do not have names,
. . these two cases represent the same division into two teams.
That is, our list has twice as many partitions than actually exist.