JEE Main 2019 (12 January)

LEVELJEE Main

Animated Solution for Mathematics - Permutations and Combinations: There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of $m$ is :

Q: Wait, why do we multiply by 2 in the formula for games between men?

Great question! The standard formula $\binom{n}{2}$ calculates the number of unique pairs in a group of $n$ people, assuming they play exactly one game. Since the problem explicitly states that each participant plays two games with every other participant, we must double the number of pairs. Thus, $2 \times \binom{n}{2}$ simplifies to $n(n-1)$.

Q: Why is the number of games between men and women $4m$?

Think of it this way: there are $m$ men and $2$ women. Each man plays each woman. That is $m \times 2$ pairings. Since each of these pairings results in $2$ games, we multiply the number of pairings by $2$. So, $2 \times (m \times 2) = 4m$. It is a direct application of the fundamental counting principle.

Select Answer:

Visualized Solution

Visualizing the Participants

Total Participants: $m$ Men and $2$ Women
Each pair plays $2$ games.

The Logic of Tournament Games

Games between $n$ people playing twice $= 2 \times (2 n) = n (n - 1)$
Games between two groups of size $n_{1}$ and $n_{2}$ playing twice $= 2 \times (n_{1} \times n_{2})$

Games Between Men

Games between Men $= 2 \times (2 m) = m (m - 1)$

Games Between Men and Women

Games between Men and Women $= 2 \times (m \times 2) = 4 m$

Setting Up the Equation

Condition: (Games between Men) $-$ (Games between Men and Women) $= 84$
Equation: $m (m - 1) - 4 m = 84$

Forming the Quadratic Equation

Expand: $m^{2} - m - 4 m = 84$
Standard Form: $m^{2} - 5 m - 84 = 0$

Factorizing the Equation

Split the middle term: $m^{2} - 12 m + 7 m - 84 = 0$
Factorize: $(m - 12) (m + 7) = 0$

Solving for $m$

Possible values: $m = 12$ or $m = - 7$
Since $m$ represents the number of men, $m > 0$ . Therefore, $m = 12$ .

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The Sigma Insight: Combinations and Selection

Solution Diagram

Analyzing the Setup

In a tournament with

n

participants where every person plays every other person twice, the total number of games is calculated by choosing

2

people out of

n

and multiplying by

2

The formula for the total number of games is:

2 \times (2 n) = 2 \times \frac{n ( n - 1 )}{2} = n (n - 1)

When considering two distinct groups of sizes

n_{1}

and

n_{2}

, the number of games played between the groups (where each person in one group plays each person in the other twice) is given by:

2 \times (n_{1} \times n_{2})

Calculating the Games

We have

m

men and

2

women. For the men playing among themselves, we apply the formula with

n = m

Games among men = m (m - 1)

Next, we calculate the games played between the

m

men and the

2

women:

Games between men and women = 2 \times (m \times 2) = 4 m

The Master Equation

The problem states that the number of games played by the men among themselves exceeds the number of games played between the men and the women by

84

. We translate this into the following algebraic equation:

m (m - 1) - 4 m = 84

Expanding the terms, we obtain:

m^{2} - m - 4 m = 84

m^{2} - 5 m - 84 = 0

Final Calculation

To solve the quadratic equation

m^{2} - 5 m - 84 = 0

, we factor the expression by finding two numbers that multiply to

- 84

and sum to

- 5

. These numbers are

- 12

and

7

The factored form is:

(m - 12) (m + 7) = 0

This yields two potential solutions:

m = 12

m = - 7

. Since the number of men must be a positive integer, we discard the negative result.

Therefore, the number of men in the tournament is $m = 12$ .

Common Traps & FAQs

Wait, why do we multiply by 2 in the formula for games between men?

Great question! The standard formula

(2 n)

calculates the number of unique pairs in a group of

n

people, assuming they play exactly one game. Since the problem explicitly states that each participant plays two games with every other participant, we must double the number of pairs. Thus,

2 \times (2 n)

simplifies to

n (n - 1)

Why is the number of games between men and women

4 m

Think of it this way: there are

m

men and

2

women. Each man plays each woman. That is

m \times 2

pairings. Since each of these pairings results in

2

games, we multiply the number of pairings by

2

. So,

2 \times (m \times 2) = 4 m

. It is a direct application of the fundamental counting principle.

Why do we reject the negative root when solving the quadratic equation?

In mathematics, quadratic equations often yield two solutions. However, in physics and combinatorics, we are dealing with physical quantities—in this case, the number of men (

m

). Since you cannot have a negative number of people in a tournament, we must discard the negative root and accept only the positive integer solution.

Select Answer:

Visualized Solution

Visualizing the Participants

The Logic of Tournament Games

Games Between Men

Games Between Men and Women

Setting Up the Equation

Forming the Quadratic Equation

Factorizing the Equation

Solving for $m$

The Sigma Insight: Combinations and Selection

Analyzing the Setup

Calculating the Games

The Master Equation

Final Calculation

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Select Answer:

Visualized Solution

Visualizing the Participants

The Logic of Tournament Games

Games Between Men

Games Between Men and Women

Setting Up the Equation

Forming the Quadratic Equation

Factorizing the Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

The Sigma Insight: Combinations and Selection

Analyzing the Setup

Calculating the Games

The Master Equation

Final Calculation

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Solving for $m$

If for some $m, n$ ; $^{6} C_{m} + 2 (^{6} C_{m + 1}) +^{6} C_{m + 2} >^{8} C_{3}$ and $^{n - 1} P_{3} :^{n} P_{4} = 1 : 8$ , then $^{n} P_{m + 1} +^{n + 1} C_{m}$ is equal to