Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ items from a set of $n$ distinct items, where order doesn't matter, is given by $\binom{n}{r} = \frac{n!}{r!(n-r)!}$. Specifically, $\binom{n}{2} = \frac{n(n-1)}{2}$ and $\binom{n}{1} = n$.
• Multiplication Principle: If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.

Step-by-Step Solution

1. Understand the Problem

• We have $m$ men and 2 women. Each participant plays two games with every other participant.
• We're given that the number of games played between men exceeds the number played between men and women by 84.
• Goal: find the value of $m$.

2. Calculate Games Played Between Men

• Goal: Determine the number of games played exclusively between the $m$ men.
• Step 2a: Find the number of pairs of men. We need to choose 2 men from the $m$ men. Order doesn't matter, so we use combinations: $\binom{m}{2} = \frac{m(m-1)}{2}$.
• Step 2b: Account for two games per pair. Each pair of men plays two games, so we multiply the number of pairs by 2: Number of games between men = $2 \times \binom{m}{2} = 2 \times \frac{m(m-1)}{2} = m(m-1)$.

3. Calculate Games Played Between Men and Women

• Goal: Determine the number of games played between men and women.
• Step 3a: Find the number of man-woman pairs. We need to choose 1 man from $m$ men and 1 woman from 2 women. The number of ways to do this is $\binom{m}{1} \times \binom{2}{1} = m \times 2 = 2m$.
• Step 3b: Account for two games per pair. Each man-woman pair plays two games, so we multiply the number of pairs by 2: Number of games between men and women = $2 \times (2m) = 4m$.

4. Formulate the Equation

• Goal: Translate the problem's condition into an equation.
• The problem states that the number of games between men exceeds the number of games between men and women by 84. Therefore: $m(m-1) - 4m = 84$.

5. Solve the Equation

• Goal: Solve the equation $m(m-1) - 4m = 84$ for $m$.
• Step 5a: Simplify the equation: $m^2 - m - 4m = 84$ $m^2 - 5m = 84$ $m^2 - 5m - 84 = 0$
• Step 5b: Factor the quadratic: $(m - 12)(m + 7) = 0$
• Step 5c: Solve for m: $m - 12 = 0$ or $m + 7 = 0$ $m = 12$ or $m = -7$

6. Validate the Solution

• Goal: Determine which solution is valid in the context of the problem.
• Since $m$ represents the number of men, it must be a positive integer. Therefore, $m = -7$ is not a valid solution.
• The only valid solution is $m = 12$.

Common Mistakes & Tips

• Carefully read the problem statement to understand the number of games played between each pair of participants.
• Remember to use combinations when the order of selection does not matter.
• Always check the validity of your solutions in the context of the problem. Negative values for quantities like the number of people are usually not valid.

Summary

This problem involved using combinations to determine the number of possible pairings between men and between men and women in a chess tournament. We then used the given information to form an equation and solve for the number of men, $m$. We found that $m=12$.

Final Answer: The final answer is \boxed{12}, which corresponds to option (A).