Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66, then the number of men who participated in the tournament lies in the interval:

Let the number of men be denoted by $$ m $$. Since there are two women, the total number of participants is $$ m + 2 $$. In the tournament, every pair of participants plays two games against each other.

First, calculate the number of games played exclusively among the men. The number of ways to choose any two men from $$ m $$ men is given by the combination formula $$ \binom{m}{2} $$, which equals $$ \frac{m(m-1)}{2} $$. Since each pair plays two games, the total number of games among the men is:

$$ 2 \times \binom{m}{2} = 2 \times \frac{m(m-1)}{2} = m(m-1) $$

Next, calculate the number of games played between the men and the women. Each man plays two games with each of the two women. Therefore, for one man, the number of games against women is $$ 2 \times 2 = 4 $$. With $$ m $$ men, the total number of such games is:

$$ m \times 4 = 4m $$

The problem states that the number of games played among the men exceeds the number of games played between the men and the women by 66. This gives the equation:

$$ m(m-1) - 4m = 66 $$

Simplify this equation step by step:

$$ m^2 - m - 4m = 66 $$

$$ m^2 - 5m = 66 $$

Bring all terms to one side to form a quadratic equation:

$$ m^2 - 5m - 66 = 0 $$

Solve this quadratic equation using factorization. We need two numbers that multiply to -66 and add to -5. The pair that satisfies this is -11 and 6, since $$ -11 \times 6 = -66 $$ and $$ -11 + 6 = -5 $$. Thus, the equation factors as:

$$ (m - 11)(m + 6) = 0 $$

Setting each factor equal to zero gives:

$$ m - 11 = 0 \quad \text{or} \quad m + 6 = 0 $$

$$ m = 11 \quad \text{or} \quad m = -6 $$

Since the number of men cannot be negative, we discard $$ m = -6 $$. Therefore, the number of men is $$ m = 11 $$.

Now, determine which interval contains 11 by examining the options:

• Option A: (11, 13] → This interval includes numbers greater than 11 and up to 13, but not 11 itself.
• Option B: (14, 17) → This interval includes numbers between 14 and 17, excluding endpoints. 11 is not in this range.
• Option C: [10, 12) → This interval includes numbers from 10 (inclusive) to 12 (exclusive). Since 10 ≤ 11 < 12, 11 is included.
• Option D: [8, 9] → This interval includes numbers from 8 to 9 inclusive. 11 is not in this range.

Hence, the number of men, 11, lies in the interval [10, 12), which corresponds to Option C.

So, the answer is Option C.