In a Chess Competition Involving Some Men and Women, Every Player Needs to Play Exactly One Game with Every Other Player GMAT Problem Solving

Question: In a chess competition involving some men and women, every player needs to play exactly one game with every other player. It was found that in 45 games, both the players were women and in 190 games, both players were men. What is the number of games in which one person was a man and other person was a woman?

40
120
180
200
220

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
The problem statement informs that:
Given:

Chess competition involves some men and women ;
Every player needs to play exactly one game with every other player. ;
In 45 games, both the players were women ;
In 190 games, both players were men. ;

Find out:

The number of games in which one person was a man and the other person was a woman.

If there are N number of people, then the formula for the number of games played
= 1+2+........+N = N∗(N−1)/2

This is because each of the n people can play chess with n - 1 people (they would not play chess with themselves), and the game between two people is not counted twice. This formula can be used for any number of people

Now, we will solve the problem statement
Let us consider the number of women as x
As stated in the question, in 45 games, both the players were women. Hence, No. of Women will be:
As per the above formula,
x∗(x−1)/2=45
This gives x=10.

Similarly, considering the number of men as y, the No. of Men:
As stated in the question, in 190 games, both players were men
Hence, y∗(y−1)/2=190
This gives y=20.

Hence, the total Men vs. Women matches or games in which one is men and the other women
=10*20
= 200.

Approach Solution 2:
The problem statement states that:
Given:

Chess competition involves some men and women ;
Every player needs to play exactly one game with every other player. ;
In 45 games, both the players were women ;
In 190 games, both players were men. ;

Find out:

The number of games in which one person was a man and the other person was a woman

Let the total number of women = x
Let the total number of men = y

As given in the question, in 45 games, both the players were women.
This implies that there are 45 ways to pick 2 women out of the total number of women:
=>xC2 = 45;
=> x!/2!(x−2)! = 45;
=>(x−1)x = 90;
=>x = 10.
As given in the question, in 190 games, both the players were men
This implies that there are 190 ways to pick 2 men out of the total number of men:
=>yC2=190;
=>y!/2!(y−2)! = 190;
=>(y−1)y = 380;
=>y = 20.
The number of games in which one person was a man and the other person was a woman = 10*20 = 200.

“In a chess competition involving some men and women, every player needs to play exactly one game with every other player”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide 2021”. The GMAT Problem Solving questions help the candidates to enhance their mathematical skills in solving quantitative problems. GMAT Quant practice papers assist the candidates to get familiar with lots of questions that will help them to score better in the exam.

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In a Chess Competition Involving Some Men and Women, Every Player Needs to Play Exactly One Game with Every Other Player GMAT Problem Solving

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