We are given that \( n \) men and 2 women participated in a chess tournament. Each player plays 2 games against every other player. Our goal is to determine the total number of players.
Step 1: Calculate the Total Games Played Among Men
The number of ways to choose 2 men from \( n \) men is: \[ \binom{n}{2} = \frac{n(n-1)}{2} \] Since each pair plays 2 games, the total number of games played among the men is: \[ G_m = 2 \times \frac{n(n-1)}{2} = n(n-1) \] Step 2: Calculate the Total Games Played Between Men and Women
Each of the \( n \) men plays 2 games against each of the 2 women. So, the total number of games played between men and women is: \[ G_{mw} = 2 \times n \times 2 = 4n \] Step 3: Use the Given Condition
We are given that the games played among men exceeded the games played between men and women by 66: \[ n(n-1) = 4n + 66 \] Step 4: Solve for \( n \)
Rearrange the equation: \[ n^2 - n = 4n + 66 \] \[ n^2 - 5n - 66 = 0 \] Solve the quadratic equation: \[ n = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-66)}}{2(1)} \] \[ n = \frac{5 \pm \sqrt{25 + 264}}{2} \] \[ n = \frac{5 \pm \sqrt{289}}{2} \] \[ n = \frac{5 \pm 17}{2} \] \[ n = \frac{5 + 17}{2} = \frac{22}{2} = 11 \quad \text{or} \quad n = \frac{5 - 17}{2} = \frac{-12}{2} = -6 \] Since \( n \) must be positive, we take \( n = 11 \).
Step 5: Find the Total Number of Players
The total number of players in the tournament is: \[ n + 2 = 11 + 2 = 13 \] Final Answer: \( {13} \)