TITLE: There are 10 different people at a party. How many ways are there to pair them off into a collection of five pairings?
QUESTION [7 upvotes]: So my thoughts are 
${10 \choose 2} * {8 \choose 2} * {6 \choose 2} * {4 \choose 2}$
if the first person , has 9 choices 
9 x 7 x 5 x 3 x 1. does this work ? 
cause i cant really make a logical thought about overlapping in chooses

REPLY [8 votes]: Isn't the double factorial (2m-1)!! the direct answer to this? For 2m people? 
Basically, each person has 9 different other persons to choose from, and after that happens, the next pair's person has 7 other people to choose from (10 minus the first pair and himself) all the way down to the last person, that has only 1 other person to choose from.
so you have 9 * 7 * 5 * 3 * 1  = (9)!! = 945