In a tournament of 9 players, each plays every other player once. How many matches are played?

The correct option is 36.

To find the total number of matches played in a tournament where each player plays every other player exactly once, we use combinations.

Since each match requires 2 players, the total number of matches is equivalent to choosing 2 players out of the total 9 players.

The formula for choosing $r$ items from $n$ items is given by:

$C(n,r)=\frac{n!}{r!(n-r)!}$

Here, $n=9$ and $r=2$.

Substituting the values into the formula:

$C(9,2)=\frac{9!}{2!(9-2)!}$

$C(9,2)=\frac{9\times 8\times 7!}{2\times 1\times 7!}$

Simplifying the expression by cancelling out $7!$:

$C(9,2)=\frac{9\times 8}{2}=\frac{72}{2}=36$

Therefore, a total of 36 matches are played in the tournament.