+678 135,135
Someone check my answer! (I can't provide a solution now and hope to later 
)
+161 @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
+26367 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
