2 [Tournaments and tournament tables]

The Russian Chess Championship is held in a single round-robin format. How many games are played if 18 chess players participate?

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The first method. Each participant must play 17 games, with two players in each game. Therefore, the total number of games is $18 \cdot 17: 2=153$.

The second method. In each game, one point is awarded. Suppose all games end in a draw. Then each participant will score $17: 2=8.5$ points. And the total number of points, and thus games, is $18 \cdot 8.5=153$.

The third method. The number of games is equal to the number of pairs of participants, which is $C_{18}^{2}$.

## Answer

153 games.