pairing possibilities in chess game
I would like to share how I ended up thinking about this.
To come up with a single instance of pairings first choose who gets to play white. There are $20\choose{10}$ ways to pick who gets to play white. Then permute the remaining 10 players, and pair them up with the white players by order (or index).
Example:
Say we randomly chose 1 3 5 7 9 11 13 15 17 19 to play white.
Then we permuted the rest and got 20 18 16 14 12 10 8 6 4 2.
Then we have the pairings:
{(1,20), (3,18), (5,16), (7,14), (9,12), (11,10), (13,8), (15,6), (17,4), (19,2)}
So we end up having $20\choose{10}$ = $\frac{20!}{10!10!}$ ways to choose who plays white, and for each those we will have $10!$ permutations of who to pair them with, this makes it so the number of pairs is $\frac{20!}{10!10!}10! = \frac{20!}{10!}$.
A set of twenty distinct items can be ordered in $20!$ ways.
A set of $10$ distinct pairs can be ordered in $10!$ ways.
A pair of distinct items can be ordered in $2!$ ways.
Since we don't care about the order of the pairs, and only care about the the order within each pair, then a set of $20$ distinct items can be subdivided into an unordered set of $10$ distinct ordered pairs in $\frac{20!}{10!}$ ways.
If it doesn't matter about the order within each pair then a set of $20$ distinct items can be subdivided into an unordered set of $10$ distinct unordered pairs in $\frac{20!}{10!\;2!^{10}}$ ways.
If we arrange the players in some order, say of height, and let the shortest player choose first, then there are 19 people for him to choose from. Then let the next shortest remaining player choose, and there are 17 people left to choose, and so on. Thus the number of ways to choose pairs is $$ 19\times17\times15\times13\times ...\times3\times1 $$ This is equal to $$ {20!\over 2^{10}10!} $$ If we care about which is white and which is black, once we have chosen the pairs, the number of ways to choose who is white and who is black among the ten pairs is $2^{10}$, so we get $$ {20!\over2^{10}10!}\times2^{10}={20!\over10!} $$