1. ## Multinomial coefficient question

Hi everyone! I don't quite understand what is the difference between these 2 questions and I was hoping if someone could explain it to me:

1. Ten children are to be divided into an A team and a B team of 5 each. The A team will play in one league and the B team in another. How many different divisions are possible? ANS: 10! / (5!5!)

I understand this solution because it's just following the formula.

2. In order to play a game of basketball, 10 children at a playground divide themselves into two teams of 5 each. How many different divisions are possible? ANS: [10! / (5!5!)] / 2!

I don't understand what the author means when he says that 'now the order of the two teams is irrelevant'. I still don't quite understand why I have to half the probability and what is the difference between this question and the one before. Can someone help me please?

2. On the first case each team will play in a different league. If we number the kids from 0 to 9:

Team A: 0-1-2-3-4 Team B: 5-6-7-8-9 (situation 1)

is different from

Team A: 5-6-7-8-9 Team B: 0-1-2-3-4 (situation 2)

Because the teams will play in different leagues. Imagine you were the kid 9. It would matter to you whether you play on Team A that goes to Division1 or Team B that goes to Division2.

On the second question you have kids on a playground ready to play a basketball game, situation 1 and situation 2 are exactly the same. The composition of the teams is equal on both situations and for a kid it's equal to play on Team A or Team B.

3. Originally Posted by deathstarx
Hi everyone! I don't quite understand what is the difference between these 2 questions and I was hoping if someone could explain it to me:

1. Ten children are to be divided into an A team and a B team of 5 each. The A team will play in one league and the B team in another. How many different divisions are possible? ANS: 10! / (5!5!)

I understand this solution because it's just following the formula.

2. In order to play a game of basketball, 10 children at a playground divide themselves into two teams of 5 each. How many different divisions are possible? ANS: [10! / (5!5!)] / 2!
This is this simple. In the first example, it makes a difference to me if I were chosen to be on team B. I would want to be on the A-team.

Under the second scenario, it is only group membership that matters.

4. Thanks! I think I get it then.