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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 :
- a)9
- b)11
- c)12
- d)7
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
There are m men and two women participating in a chess tournament. Eac...
Let m-men, 2-women
m2 – 5m – 84 = 0 ⇒ (m – 12) (m + 7) = 0 m = 12
Most Upvoted Answer
There are m men and two women participating in a chess tournament. Eac...
Let's assume that there are m men participating in the chess tournament. Since each participant plays two games with every other participant, the total number of games played by the men can be calculated using the combination formula:
Total games played by men = C(m, 2) = m(m-1)/2
Similarly, the number of games played between the men and the women can be calculated using the combination formula:
Total games played between men and women = C(m, 1) * C(2, 1) = m * 2 = 2m
According to the given information, the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84. Mathematically, this can be represented as:
m(m-1)/2 - 2m = 84
Now, let's solve this equation to find the value of m.
Simplifying the equation:
m(m-1)/2 - 2m = 84
(m^2 - m)/2 - 2m = 84
m^2 - m - 4m - 168 = 0
m^2 - 5m - 168 = 0
Now, we can factorize the quadratic equation:
(m - 12)(m + 14) = 0
From the equation, we can see that either m - 12 = 0 or m + 14 = 0. So, the possible values of m are:
m = 12 or m = -14
Since the number of participants cannot be negative, the value of m must be 12.
Therefore, the value of m is 12. So, the correct answer is option C.
Total games played by men = C(m, 2) = m(m-1)/2
Similarly, the number of games played between the men and the women can be calculated using the combination formula:
Total games played between men and women = C(m, 1) * C(2, 1) = m * 2 = 2m
According to the given information, the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84. Mathematically, this can be represented as:
m(m-1)/2 - 2m = 84
Now, let's solve this equation to find the value of m.
Simplifying the equation:
m(m-1)/2 - 2m = 84
(m^2 - m)/2 - 2m = 84
m^2 - m - 4m - 168 = 0
m^2 - 5m - 168 = 0
Now, we can factorize the quadratic equation:
(m - 12)(m + 14) = 0
From the equation, we can see that either m - 12 = 0 or m + 14 = 0. So, the possible values of m are:
m = 12 or m = -14
Since the number of participants cannot be negative, the value of m must be 12.
Therefore, the value of m is 12. So, the correct answer is option C.
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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 :a)9b)11c)12d)7Correct answer is option 'C'. Can you explain this answer?
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