1. Interpret the situation

There are $m$ men and $2$ women, so total participants are $m+2$.

Each pair of participants plays 2 games with each other.

We must compare:

• games played by the men among themselves
• games played between men and women

It is given that: $\text{(games among men)} - \text{(games between men and women)} = 84$

2. Number of games among the men

Number of pairs of men: $\binom{m}{2}$

Since each pair plays 2 games, total games among men: $2\binom{m}{2} = 2\cdot \frac{m(m-1)}{2} = m(m-1)$

3. Number of games between men and women

Each of the $m$ men plays with each of the $2$ women.

So number of man-woman pairs: $m\cdot 2 = 2m$

Each such pair plays 2 games, so total games between men and women: $2(2m)=4m$

4. Use the given condition

$m(m-1) - 4m = 84$

Simplify: $m^2 - m - 4m = 84$ $m^2 - 5m = 84$ $m^2 - 5m - 84 = 0$

Factorize: $m^2 - 12m + 7m - 84 = 0$ $m(m-12) + 7(m-12)=0$ $(m-12)(m+7)=0$

So, $m=12 \quad \text{or} \quad m=-7$

Since number of men cannot be negative, $m=12$

5. Check options

• A: $12$ ✅
• B: $9$ ❌
• C: $7$ ❌
• D: $11$ ❌

Hence the correct option is A.