Key Concepts and Formulas

• Fundamental Principle of Counting (Multiplication Principle): If one event can occur in $m$ ways and a second event can occur in $n$ ways, and the events are independent, then the number of ways the two events can occur together is $m \times n$.
• Combination Formula: The number of ways to choose $r$ objects from a set of $k$ distinct objects is given by ${}^k{C_r} = \frac{k!}{r!(k-r)!}$. In the special case where $r=1$, ${}^k{C_1} = k$.

Step-by-Step Solution

1. Understand the Problem

We are given two teams, A and B, with known numbers of boys and girls in each. We are also given the total number of possible single matches (boy vs. boy and girl vs. girl) between the teams. The goal is to find the number of girls, $n$, in Team A.

2. Calculate the Number of Boy-Boy Matches

To form a boy-boy match, we need to choose one boy from Team A and one boy from Team B.

• Number of ways to choose 1 boy from Team A: Team A has 7 boys. The number of ways to choose 1 boy is ${}^7{C_1} = 7$
• Number of ways to choose 1 boy from Team B: Team B has 4 boys. The number of ways to choose 1 boy is ${}^4{C_1} = 4$

Using the multiplication principle, the total number of boy-boy matches is: $\text{Total boy-boy matches} = {}^7{C_1} \times {}^4{C_1} = 7 \times 4 = 28$ Explanation: Each of the 7 boys in Team A can play against any of the 4 boys in Team B, resulting in $7 \times 4 = 28$ possible matches.

3. Calculate the Number of Girl-Girl Matches

To form a girl-girl match, we need to choose one girl from Team A and one girl from Team B.

• Number of ways to choose 1 girl from Team A: Team A has $n$ girls. The number of ways to choose 1 girl is ${}^n{C_1} = n$
• Number of ways to choose 1 girl from Team B: Team B has 6 girls. The number of ways to choose 1 girl is ${}^6{C_1} = 6$

Using the multiplication principle, the total number of girl-girl matches is: $\text{Total girl-girl matches} = {}^n{C_1} \times {}^6{C_1} = n \times 6 = 6n$ Explanation: Each of the $n$ girls in Team A can play against any of the 6 girls in Team B, resulting in $n \times 6 = 6n$ possible matches.

4. Formulate the Equation and Solve for $n$

The total number of matches is the sum of the boy-boy matches and the girl-girl matches, which is given as 52. $\text{Total matches} = \text{Total boy-boy matches} + \text{Total girl-girl matches}$ Substituting the values we calculated: $52 = 28 + 6n$ Now, we solve for $n$: $52 - 28 = 6n$ $24 = 6n$ $n = \frac{24}{6}$ $n = 4$

5. Verify the Solution

If $n=4$, then there are $6n = 6(4) = 24$ girl-girl matches. Adding the 28 boy-boy matches, we get a total of $28 + 24 = 52$ matches, which matches the given information.

Common Mistakes & Tips

• Distinguishing Multiplication and Addition: Ensure you multiply the number of ways to choose a boy/girl from each team (independent events) and add the number of boy-boy and girl-girl matches since they are mutually exclusive.
• Using the Correct Combination Formula: While the combination formula is used, remember that ${}^k{C_1} = k$, which simplifies the calculations.

Summary

The problem requires us to find the number of girls in Team A by calculating the possible boy-boy and girl-girl matches between the two teams. We use the Fundamental Principle of Counting to find the number of each type of match and then set up an equation based on the total number of matches given. Solving the equation, we find that there are 4 girls in Team A.

The final answer is \boxed{4}, which corresponds to option (C).