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# can someone help?

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147
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How many ways are there to split 14 people into seven pairs?

Jul 9, 2020

#1
+549
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135,135

Someone check my answer! (I can't provide a solution now and hope to later )

Jul 9, 2020
#2
+161
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@amazingxin777 I think it's right. 14 choose 2 * 12 choose 2 * 10 choose 2 * 8 choose 2 * 6 choose 2 * 4 choose 2 * 2 choose 2.

Then, you have to divide out the ways to order the pairs (Since A+B, C+D is the same pairing as C+D, A+B). There are 7!=5040 ways to order the 7 pairs, so that leaves 135,135 total pairings.

- tigernathan

Jul 9, 2020
#3
+25569
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How many ways are there to split 14 people into seven pairs?

The first person of the 14 people can form a pair with $${\color{red}13}$$ other people. We now have one pair less.
The first person of the remaining 12 people can form a pair with $${\color{red}11}$$ other people. We now have two pairs less.
The first person of the remaining 10 people can form a pair with $${\color{red}9}$$ other people. We now have three pairs less.
The first person of the remaining 8 people can form a pair with $${\color{red}7}$$ other people. We now have four pairs less.
The first person of the remaining 6 people can form a pair with $${\color{red}5}$$ other people. We now have five pairs less.
The first person of the remaining 4 people can form a pair with $${\color{red}3}$$ other people. We now have six pairs less.
The first person of the remaining 2 people can form a pair with $${\color{red}1}$$ other people. We now have seven pairs less.

There are $${\color{red}1}*{\color{red}3}*{\color{red}5}*{\color{red}7}*{\color{red}9}*{\color{red}11}*{\color{red}13} = \mathbf{135135}$$ ways to split 14 people into seven pairs

Jul 10, 2020
edited by heureka  Jul 10, 2020