Serena and Venus were only two women participiting in a chess tournament. Every participant played two games with every other participant. The number of games that men played between themselves proved to exceed by 66, compared to the number of games the men played with women. How many participants were there?

Options

A. 156
B. 610
C. 13
D. 108

Answer

To solve the problem step by step, we will first define the variables and then set up the equations based on the information provided.

### Step 1: Define the Variables
Let \( X \) be the number of men participating in the chess tournament. Since there are 2 women (Serena and Venus), the total number of participants will be \( X + 2 \).

### Step 2: Calculate Games Played Between Men
Each pair of men plays 2 games against each other. The number of ways to choose 2 men from \( X \) men is given by the combination formula \( \binom{X}{2} \), which is calculated as:
\[ \binom{X}{2} = \frac{X(X-1)}{2} \]
Thus, the total number of games played between men is:
\[ \text{Games between men} = 2 \times \binom{X}{2} = 2 \times \frac{X(X-1)}{2} = X(X-1) \]

### Step 3: Calculate Games Played Between Men and Women
Each man plays 2 games with each woman. Since there are 2 women, the total number of games played between men and women is:
\[ \text{Games with women} = 2 \times X \times 2 = 4X \]

### Step 4: Set Up the Equation
According to the problem, the number of games that men played between themselves exceeds the number of games that men played with women by 66. Therefore, we can set up the equation:
\[ X(X-1) = 4X + 66 \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ X(X-1) - 4X - 66 = 0 \]
This simplifies to:
\[ X^2 - 5X - 66 = 0 \]

### Step 6: Solve the Quadratic Equation
We can solve the quadratic equation using the quadratic formula:
\[ X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -5, c = -66 \). Plugging in these values:
\[ X = \frac{5 \pm \sqrt{(-5)^2 - 4 \times 1 \times (-66)}}{2 \times 1} \]
\[ X = \frac{5 \pm \sqrt{25 + 264}}{2} \]
\[ X = \frac{5 \pm \sqrt{289}}{2} \]
\[ X = \frac{5 \pm 17}{2} \]
Calculating the two possible values:
1. \( X = \frac{22}{2} = 11 \)
2. \( X = \frac{-12}{2} = -6 \) (not a valid solution)

Thus, \( X = 11 \).

### Step 7: Calculate Total Participants
The total number of participants is:
\[ \text{Total participants} = X + 2 = 11 + 2 = 13 \]

### Final Answer
The total number of participants in the tournament is **13**.

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