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In a class of 15 students a quiz is held. The sum of their scores is 100. How many students atleast must have the same score?
Correct answer is '2'. Can you explain this answer?
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In a class of 15 students a quiz is held. The sum of their scores is 1...
Suppose the scores are all different.
Therefore, we can arrange them in order s1< s2< ...< s15. The smallest possible values here would be s1=0, s2=1, s3=2, ..., s15=14.
If we add these scores, we get (14)(15)/2 = 105. This is a contradiction since the scores are supposed to sum to 100.
Thus, the scores cannot be all different. Atleast 2 students must have the same score.
Most Upvoted Answer
In a class of 15 students a quiz is held. The sum of their scores is 1...
Understanding the Problem
In a class of 15 students, the total score from a quiz is 100. To determine how many students must have the same score, we can use the concept of averages and the Pigeonhole Principle.
Average Score Calculation
- The average score for the students can be calculated as follows:
- Average = Total Score / Number of Students
- Average = 100 / 15 ≈ 6.67
This indicates that if all students had different scores, the scores would need to be distributed around this average.
Pigeonhole Principle
- The Pigeonhole Principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.
- Here, we can consider the possible distinct scores as "containers" and the students as "items".
Score Distribution Implications
- If we assume the scores are whole numbers, the lowest possible score is 0, and the highest would be 10 (since 10 students scoring 10 would sum to 100).
- Possible distinct scores could be from 0 to 10, providing us with 11 possible scores.
Conclusion
- Since we have 15 students and only 11 possible distinct scores, by the Pigeonhole Principle, at least:
- (15 students) / (11 distinct scores) = 1.36 scores per score category
- This means at least 2 students must share the same score, as we cannot assign all 15 students unique scores without exceeding the available score options.
Thus, the answer is that at least 2 students must have the same score.
In a class of 15 students, the total score from a quiz is 100. To determine how many students must have the same score, we can use the concept of averages and the Pigeonhole Principle.
Average Score Calculation
- The average score for the students can be calculated as follows:
- Average = Total Score / Number of Students
- Average = 100 / 15 ≈ 6.67
This indicates that if all students had different scores, the scores would need to be distributed around this average.
Pigeonhole Principle
- The Pigeonhole Principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.
- Here, we can consider the possible distinct scores as "containers" and the students as "items".
Score Distribution Implications
- If we assume the scores are whole numbers, the lowest possible score is 0, and the highest would be 10 (since 10 students scoring 10 would sum to 100).
- Possible distinct scores could be from 0 to 10, providing us with 11 possible scores.
Conclusion
- Since we have 15 students and only 11 possible distinct scores, by the Pigeonhole Principle, at least:
- (15 students) / (11 distinct scores) = 1.36 scores per score category
- This means at least 2 students must share the same score, as we cannot assign all 15 students unique scores without exceeding the available score options.
Thus, the answer is that at least 2 students must have the same score.
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In a class of 15 students a quiz is held. The sum of their scores is 100. How many students atleast must have the same score?Correct answer is '2'. Can you explain this answer?
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