Line up the 20 players in some order then say the first two are a pair, the next two are a pair, etc. This overcounts by a factor of 10! because we don’t care about the order of the games. So in total we have
ways for them to play. This correctly counts for the whether player A plays white or black. If we didn’t care we would need to divide by 2.
Another way to look at it is to choose the 10 players who will play white then let each of them choose their opponent from the other 10 players. This gives a total of
possibilities of how they are matched up. We don’t care about the order of the players who play white but once we’ve chosen them the order of the players who play black matters since different orders mean different pairings.