## Task 2 - 270622

(a) In a competition, exactly four teams $A, B, C$, and $D$ participated, and each of these teams played exactly one game against each of the other teams. List these games!

(b) In another competition, each of the participating teams played exactly one game against each of the other participating teams, resulting in exactly 21 games.

How many teams participated in this competition in total?

Show that with the number of teams you have determined, exactly 21 games took place!
(a) The games (denoted by writing the letters consecutively) are $A B, A C, A D, B C, B D$, $C D$.

(b) If a fifth team $E$ is added, then in addition to these six games, there are four more $(A E, B E, C E, D E)$, making a total of ten games.

If a sixth team is added, then in addition to the existing games, there are exactly five more games that it has to play against $A, B, C, D, E$, making a total of 15 games.

A seventh team has to play exactly six more games. Thus, it is shown that with seven teams, exactly 21 games will take place.