At the first meeting of the chess club, the 10 members decided that everyone in the club should play everyone else one time to determine the best players for the inter-school competitions. How many games had to be played?

Answer 1

Same as the standard "hand-shake" problem
number of games = 10 choose 2
= C(10,2) = 45

In the hand-shake problem, we have 10 people and they all shake hands on meeting.
so each of the 10 people can shake hands with 9 others for 90 shakes
but that would include A shaking with D and D shaking with A, the same handshake, so we divide our 90 by 2 to eliminate all duplicate pairs.
The same argument applies to your players
Player A vs B is the same game as B vs A