Problem 21 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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: (a) \([8,9]\) (b) \([10,12)\) (c) \((11,13]\) (d) \((14,17)\)

Step-by-Step Solution

Answer

The number of men lies in the interval (11,13].

Step 1: Understanding the Problem

In the chess tournament, every participant played two games with each of the other participants. There are some men (let's denote the number of men as \( n \)) and two women. We need to determine the interval containing \( n \) based on the information about the games played between men and between men and women.

Step 2: Calculating Total Games Played Among Men

If there are \( n \) men, then the number of games played among the men is given by the formula for combinations:\[2 \times \binom{n}{2} = n(n-1)\]This accounts for two games played between every pair of men.

Step 3: Calculating Games Played Between Men and Women

Each man plays two games with each woman. Since there are 2 women, the total number of games played between men and women is:\[2 \times 2 \times n = 4n\]

Step 4: Setting up the Equation

According to the problem, the games played among men exceed the games played with women by 66 games:\[n(n-1) = 4n + 66\]

Step 5: Simplifying the Equation

Solve the equation for \( n \):\[ n^2 - n = 4n + 66\]Simplifying:\[ n^2 - 5n - 66 = 0\]This is a quadratic equation in terms of \( n \).

Step 6: Solving the Quadratic Equation

Use the quadratic formula to solve for \( n \):\[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a = 1, b = -5, c = -66 \):\[n = \frac{5 \pm \sqrt{25 + 264}}{2}\]\[n = \frac{5 \pm \sqrt{289}}{2}\]\[n = \frac{5 \pm 17}{2}\]The possible values for \( n \) are:\[n = \frac{22}{2} = 11\] and \[ n = \frac{-12}{2} = -6\]Since the number of participants can't be negative, \( n = 11 \).

Step 7: Verifying the Interval Choice

The correct interval containing \( n = 11 \) is \((11,13]\). Therefore, the answer is (c).

Key Concepts

Combinatorial Mathematics

Combinatorial mathematics is a field of mathematics primarily concerned with counting, arranging, and structuring. In the context of the chess tournament problem, combinatorial analysis helps determine the number of games played amongst participants.

In tournaments, each player meets others in matches, and we need to know how many unique pairs of players can be formed. This is where combinations come into play. The formula for combinations, given as \( \binom{n}{2} \), provides the count of pairs when choosing 2 participants from \( n \) individuals. Here, it calculates how many pairs of men can be formed for games.

Since each pair plays two games, multiplying the combinations by 2 gives us the total number of games played among men alone, expressed as \( 2 \times \binom{n}{2} = n(n-1) \). This simple use of combinatorial mathematics illustrates its power in analyzing tournament structures efficiently.

Quadratic Equations

Quadratic equations are vital tools in finding unknown values when dealing with various problems, like the one in this tournament scenario. A standard quadratic equation is of the form \( ax^2 + bx + c = 0 \). In our chess tournament problem, after setting up the relational equation of games, simplifying it led to a quadratic equation: \( n^2 - 5n - 66 = 0 \).

Solving these equations often requires the quadratic formula, \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This helps us find the possible values for \( n \), which represents the number of men participating. Using this formula, we substitute \( a = 1 \), \( b = -5 \), and \( c = -66 \), which after calculation, yields two potential solutions for \( n \).

Importantly, quadratic equations can yield negative results, like \( n = -6 \) here, but in real situations like this where participants must be a positive count, these solutions are discarded.

Problem-Solving in Mathematics

Effective problem-solving in mathematics involves multiple steps: understanding the problem, formulating equations, and logically deducing solutions. The chess tournament challenge demonstrates this process well.

First, the problem is thoroughly understood. Participants and games between them are identified. Formulating the equations based on combinatorial mathematics grounds the problem in arithmetic, leading to a meaningful structure for advancing towards an answer.

Once the quadratic equation is derived, solving it involves applying mathematical skills, such as using the quadratic formula. This systematic approach ensures that mathematical reasoning remains consistent and reliable throughout the process. Additionally, solution verification against given conditions, like ensuring \( n = 11 \) falls in the correct interval \((11,13]\), reinforces a solid mathematical solution.