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In how many ways can $6$ children be divided into $3$ pairs if the order of pairs does not matter?

In my book, the answer is only :

$$\frac{^6C_2\cdot^4C_2\cdot^2C_2}{3!}=\frac{1}{6}\cdot90=15$$

I do not understand why we divide it by $3!$ . What's the intuition behind this?

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    $\begingroup$ ${6\choose 2}$ are the number of pairs. Once you pick a first pair on a list there are ${4\choose 2}$ pairs to put in the second position only a list and once you pick that there is ${2\choose 2}$ (0r $1$) pair to put in the last place on the list. SO there are ${6\choose 2}{4\choose 2}{2\choose 2}$ ways to make a list of three pairs. But the order of the list does not matter. So you divide it by the number of ways to order three items on a list. $\endgroup$
    – fleablood
    Commented Apr 9, 2019 at 23:00

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Cause you are choosing an order in the groups. If i have the children as $x_1x_2\dots x_6$ then if we pick $2$(by the $\binom{6}{2}$) we can color it with $\color{red}{red}$ for example $$\color{red}{x_1}x_2\color{red}{x_3}x_4x_5x_6,$$ then we pick other two and color them with blue, for example $$\color{red}{x_1}\color{blue}{x_2}\color{red}{x_3}x_4\color{blue}{x_5}x_6.$$ This is equivalent to $$\color{blue}{x_1}\color{red}{x_2}\color{blue}{x_3}x_4\color{red}{x_5}x_6$$ and any permutation of the $3$ colors. To avoid this you divide by the $3!$order of the colors.

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Was going through a similar question and found a visual aid that helped me picture the question. Imagine a 6x6 table, where it is labeled 1 thru 6 horizontally and vertically as such.

enter image description here

Next we pair them up:

enter image description here

Since we cannot pair two of the same person, hence, we will take out 11, 22, ... 66; which are the diagonal pairs highlighted in yellow

enter image description here

Lastly, we take out half the triangle which double counted AB, BA, etc... highlighted in green. This also interprets 2 ways (e.g.: AB or BA) of choosing. And yes this is pascals triangle. This visual aid is only applicable when there is equal distribution.

Mathematically; we start with 6 x 6 = 36, next we subtract the yellow highlighted pairs 36 - 6 = 30. Lastly we subtract the green highlighted pairs 30 - 15 = 15.

enter image description here

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