There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then teh value of m is

Options

A. 9
B. 11
C. 12
D. 7

Answer

To solve the problem, we need to find the value of \( m \) given the conditions of the chess tournament. Let's break it down step by step.

### Step 1: Calculate the number of games played among men
The number of ways to choose 2 men from \( m \) men is given by the combination formula \( \binom{m}{2} \). Each pair of men plays 2 games against each other. Therefore, the total number of games played among the men is:

\[ \text{Games among men} = 2 \cdot \binom{m}{2} = 2 \cdot \frac{m(m-1)}{2} = m(m-1) \]

### Step 2: Calculate the number of games played between men and women
Each man plays 2 games against each woman. Since there are 2 women, the total number of games played between the men and the women is:

\[ \text{Games between men and women} = 2 \cdot m \cdot 2 = 4m \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the number of games played among the men exceeds the number of games played between the men and the women by 84. This can be expressed as:

\[ m(m-1) = 4m + 84 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:

\[ m(m-1) - 4m - 84 = 0 \]

This simplifies to:

\[ m^2 - m - 4m - 84 = 0 \]

\[ m^2 - 5m - 84 = 0 \]

### Step 5: Solve the quadratic equation
Now we can factor the quadratic equation:

\[ m^2 - 5m - 84 = 0 \]

To factor, we look for two numbers that multiply to \(-84\) and add to \(-5\). These numbers are \(-12\) and \(7\). Thus, we can write:

\[ (m - 12)(m + 7) = 0 \]

### Step 6: Find the possible values for \( m \)
Setting each factor to zero gives:

\[ m - 12 = 0 \quad \Rightarrow \quad m = 12 \]
\[ m + 7 = 0 \quad \Rightarrow \quad m = -7 \quad (\text{not valid since } m \text{ must be positive}) \]

Thus, the only valid solution is:

\[ m = 12 \]

### Final Answer
The value of \( m \) is \( 12 \).

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