# Ordered Partition

• Dec 31st 2009, 01:58 PM
novice
Ordered Partition
I have this math problem that drives me absolutely nuts. I saw this same one at high school and again in college. Here it is:

In how many ways can 9 students be divided into three teams?

The answer to this question is ${8\choose2}{5\choose 2}=280$

At high school, I asked my teacher why the answer being it is. He asked me to consider myself a coach. By so, I appoint a student for a team captain for each team and have each team captain choose two others for her team. When I said, "If I were the coach, why can't I choose them myself?" He was speechless.

Now I see this same problem again in a different book that gave the answer in this form: $\frac{9!}{3!3!3!}\cdot\frac{1}{3!}=280$

I understand this part: $\frac{9!}{3!3!3!}$, but I don't understand why it's multiplied by $\frac{1}{3!}$

Can anyone explain?
• Dec 31st 2009, 02:15 PM
Plato
If you were to gave a RED team, a BLUE team and a GREEN team then $\frac{9!}{(3!)^3}$ would be correct.
That is because the color team can make a difference. I don’t want to be on the BLUE team.

But because there is nothing to distinguish the teams we divide by another $3!$.

If you want to divide twenty people into four groups of five each the number is $\frac{20!}{(5!)^4(4!)}$.
• Dec 31st 2009, 03:40 PM
novice
Quote:

Originally Posted by Plato
If you were to gave a RED team, a BLUE team and a GREEN team then $\frac{9!}{(3!)^3}$ would be correct.
That is because the color team can make a difference. I don’t want to be on the BLUE team.

But because there is nothing to distinguish the teams we divide by another $3!$.

If you want to divide twenty people into four groups of five each the number is $\frac{20!}{(5!)^4(4!)}$.

Is $3!$ in the denominator meant to get rid of the order:
{RED, BLUE, GREEN} $,${RED, GREEN, BLUE} $,${GREEN, BLUE, RED} $,${GREEN, RED, BLUE} $,${BLUE, GREEN, RED} $,${BLUE, RED, GREEN}=3!
• Dec 31st 2009, 04:01 PM
Plato
Quote:

Originally Posted by novice
Is $3!$ in the denominator meant to get rid of the order:

Yes, in general.
Study the other example I gave you.
• Dec 31st 2009, 04:08 PM
novice
Quote:

Originally Posted by Plato
Yes, in general.
Study the other example I gave you.

Thank you, Plato,
Now I don't have to get up in the middle of the night banging my head.