Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single

Q: Wait, why do we multiply the number of boys in Team A by the number of boys in Team B? Shouldn't we add them?

Great question! Think of it this way: for every single boy in Team A, he has to play against every boy in Team B. If there are 7 boys in Team A, the first boy plays 4 matches, the second boy plays 4 matches, and so on, for all 7 boys. That is $4 + 4 + 4 + 4 + 4 + 4 + 4$, which is exactly $7 \times 4$. Multiplication is just repeated addition!

Q: Why are there no mixed matches? The problem doesn't explicitly say 'no mixed matches' in a negative way.

In competitive combinatorics problems, when the rule is stated as 'a boy plays against a boy and a girl plays against a girl,' it defines the entire sample space of allowed matches. Any other combination (like a boy playing a girl) is excluded by the definition of the match criteria provided.

Q: What if the calculation for $n$ resulted in a fraction? Would that be valid?

That is a fantastic intuition check! Since $n$ represents the number of girls in a team, it must be a non-negative integer. If your algebra ever leads to a non-integer result for a count of people, it is a clear signal to go back and re-read the problem or check your arithmetic for a calculation error.

Question

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

Options:

(A) 5

(B) 2

(C) 4

(D) 6

Sigma Percentile · Accepted Answer

### The Elegance of Combinatorial Logic

Welcome, future engineer! Today, we are going to dissect a problem that might seem like a simple counting exercise, but it is actually a beautiful lesson in the **Fundamental Principle of Counting**.

When you face problems like this in the JEE Advanced, the key is to strip away the narrative and look at the underlying structure. Let us break this down together.

### Phase 1: Visualizing the Teams

Imagine you are the coach of two teams, Team A and Team B. You have a clipboard, and you need to organize a series of matches.

Team A is a bit of a mystery; you know they have $7$ boys, but the number of girls, which we will call $n$, is unknown. Team B is fully scouted: they have $4$ boys and $6$ girls.

Our goal is to find $n$ such that the total number of single matches equals $52$.

### Phase 2: The Logic of Matches

The rule is strict: a boy plays only against a boy, and a girl plays only against a girl. This is a classic case of independent events.

We are not dealing with a chaotic mix of players. Instead, we have two separate, non-interacting pools of matches. The total number of matches is the sum of the matches in the 'boys' pool and the matches in the 'girls' pool.

Mathematically, we can write this as:

$$	ext{Total Matches} = (	ext{Boys in A} 	imes 	ext{Boys in B}) + (	ext{Girls in A} 	imes 	ext{Girls in B})$$

### Phase 3: The Calculation

Let us calculate the boy-versus-boy matches first. We have $7$ boys in Team A and $4$ boys in Team B.

Since every boy in Team A must play every boy in Team B, we use the multiplication principle. For each of the $7$ boys in Team A, there are $4$ potential opponents in Team B.

Therefore, the total number of boy matches is $7 	imes 4 = 28$. This part of the equation is rock solid; it does not change regardless of how many girls are on the team.

Now, let us turn to the girls. We have $n$ girls in Team A and $6$ girls in Team B. Using the same logic, the number of girl-versus-girl matches is $n 	imes 6$, or $6n$.

### Phase 4: Solving for the Unknown

We are told the grand total of matches is $52$. Now, we simply assemble our pieces into a linear equation:

$$28 + 6n = 52$$

This is where many students rush and make a sign error. Take a breath. We need to isolate $6n$ by subtracting $28$ from both sides:

$$6n = 52 - 28$$

$$6n = 24$$

Finally, we divide by $6$ to find our answer:

$$n = \frac{24}{6} = 4$$

### The Takeaway

And there you have it! The number of girls in Team A is **$4$**.

It is a simple result, but notice the process: we identified the constraints, separated the problem into independent components, applied the counting principle, and solved the resulting equation. This is the exact mindset you need for the JEE.

Never let a word problem intimidate you; just translate it into the language of mathematics, and the answer will reveal itself.

Animated Solution for Mathematics - Permutations and Combinations: Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

Select Answer:

Visualized Solution

Visualizing the Teams

Understanding the Match Rule

Calculating Boy-vs-Boy Matches

Total Boy Matches

Calculating Girl-vs-Girl Matches

Setting up the Total Equation

Isolating the Variable

Solving the Subtraction

Finding the Final Value of $n$

The Sigma Insight: Fundamental Principle of Counting

The Elegance of Combinatorial Logic

Phase 1

Phase 2

Phase 3

Phase 4

The Takeaway

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