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There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants is
- a)6
- b)11
- c)13
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
There were two women participating in a chess tournament. Every partic...
Let there be n men participants. Then, the number of games that the men play between themselves is 2. nC2 and the number of games that the men played with the women is 2.(2n)
∴ 2nC2 − 2⋅2n = 66 (given)
⇒ n (n−1) − 4n − 66 = 0
⇒ n2 − 5n − 66 = 0
⇒(n + 5) (n − 11) = 0
⇒ n = 11
∴ Number of participants =11 men+2 women=13
∴ 2nC2 − 2⋅2n = 66 (given)
⇒ n (n−1) − 4n − 66 = 0
⇒ n2 − 5n − 66 = 0
⇒(n + 5) (n − 11) = 0
⇒ n = 11
∴ Number of participants =11 men+2 women=13
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There were two women participating in a chess tournament. Every partic...
Given information:
- There were two women participating in a chess tournament.
- Every participant played two games with the other participants.
- The number of games that the men played between themselves exceeded by 66 the number of games that the men played with the women.
Let's assume:
- The number of men participating in the tournament is 'x'.
- The total number of participants, including men and women, is 'y'.
Number of games played:
- Each participant plays two games with every other participant.
- The total number of games played will be the sum of all pairwise combinations of participants.
Number of games played between men:
- Since there are 'x' men participating, the number of games played between men can be calculated using the combination formula: C(x, 2) = x(x-1)/2.
Number of games played between men and women:
- Since there are 'x' men and 2 women participating, the number of games played between men and women can be calculated using the multiplication principle as x * 2.
Given condition:
- The number of games played between men exceeds by 66 the number of games played between men and women.
- Mathematically, it can be represented as: x(x-1)/2 = x * 2 + 66.
Solving the equation:
- Simplifying the equation, we get x(x-1)/2 - x * 2 = 66.
- Expanding the equation, we get x^2 - x - 4x = 132.
- Combining like terms, we get x^2 - 5x - 132 = 0.
- Factoring the quadratic equation, we get (x - 11)(x + 12) = 0.
- Solving for 'x', we get x = 11.
Number of participants:
- The total number of participants, including men and women, is 'y'.
- Since there are 2 women participating, the total number of participants will be 11 (x) + 2 = 13.
Conclusion:
The correct answer is option 'C', i.e., 13 participants.
- There were two women participating in a chess tournament.
- Every participant played two games with the other participants.
- The number of games that the men played between themselves exceeded by 66 the number of games that the men played with the women.
Let's assume:
- The number of men participating in the tournament is 'x'.
- The total number of participants, including men and women, is 'y'.
Number of games played:
- Each participant plays two games with every other participant.
- The total number of games played will be the sum of all pairwise combinations of participants.
Number of games played between men:
- Since there are 'x' men participating, the number of games played between men can be calculated using the combination formula: C(x, 2) = x(x-1)/2.
Number of games played between men and women:
- Since there are 'x' men and 2 women participating, the number of games played between men and women can be calculated using the multiplication principle as x * 2.
Given condition:
- The number of games played between men exceeds by 66 the number of games played between men and women.
- Mathematically, it can be represented as: x(x-1)/2 = x * 2 + 66.
Solving the equation:
- Simplifying the equation, we get x(x-1)/2 - x * 2 = 66.
- Expanding the equation, we get x^2 - x - 4x = 132.
- Combining like terms, we get x^2 - 5x - 132 = 0.
- Factoring the quadratic equation, we get (x - 11)(x + 12) = 0.
- Solving for 'x', we get x = 11.
Number of participants:
- The total number of participants, including men and women, is 'y'.
- Since there are 2 women participating, the total number of participants will be 11 (x) + 2 = 13.
Conclusion:
The correct answer is option 'C', i.e., 13 participants.
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There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66the number of games that the men played with the women. The number of participants isa)6b)11c)13d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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