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How many ways are there to split a dozen people into 3 teams, where one team has 2 people, and the other two teams have 5 people each?

I tried to answer the question and got the answer as $\binom {12}{2} \binom{10}{5}$.

But the solution I found out to be is $\frac{\binom {12}{2} \binom{10}{5}}{2!}$. 2! it seems was because I was forming two groups of 5 people.

So I thought of a simpler problem.

If I have four elements {A,B,C,D} and I want to split them in groups of two So there would $\binom{4}{2} =6 $ ways of doing it.{A,B},{A,C}{A,D}{B,C}{B,D}{C,D}.But by the above logic I would have to divide the $\binom {4}{2}$ by 2!. But this doesn't seem correct.

Where is my understanding going wrong?

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  • $\begingroup$ It is correct to divide by $2!=2$ in the simpler problem, because if you think about it, $\{A,B\}$ and $\{C,D\}$(for example) give the same result of $\{\{A,B\},\{C,D\}\}$. We can form these pairs with all the possible sets, so it is correct to divide by $2$ in that situation. But if order does matter for the problem, then you don't divide by $2$, because then $\{\{A,B\},\{C,D\}\}$ and $\{\{C,D\},\{A,B\}\}$ would be different combinations, so we can't take out half the sets. $\endgroup$
    – Aiden Chow
    Oct 19, 2020 at 7:37

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You want to split your 4 elements into TWO groups of 2:

$(A,B)$ and $(C,D)$

$(A,C)$ and $(B,D)$

$(A,D)$ and $(B,C)$

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  • $\begingroup$ So $\binom {4}{2}$ is just the no of ways of choosing two people from a group of 4. where as to find the no of ways in which two groups of two can found I divide by 2!. In the case of the earlier problem I was forming two groups of 5 people hence divide by 2!. So If I have n objects and wants to form k groups with r elements in each group I use $\frac {\binom{n}{r}}{k!}$ $\endgroup$
    – Orpheus
    Oct 19, 2020 at 7:52
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    $\begingroup$ Not exactly. Suppose you want to form $k$ groups all of size $r$ then the number of ways you can do this is counted as $$\frac{\binom{n}{r}\binom{n-r}{r}\ldots \binom{n-kr}{r}}{k!}$$ $\endgroup$ Oct 20, 2020 at 15:39

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