Exactly 3000 students wrote a multiple-choice test comprising exactly five questions, each question, in turn, having exactly five answer choices, exactly one of which is correct. Every question correctly answered fetches 4 marks and every question incorrectly answered fetches –1 mark. If each student attempted all the questions and the answer choices marked by no two students are the same for all the five questions, then the number of students with a net positive score in the test is at least
Question
Exactly 3000 students wrote a multiple-choice test comprising exactly five questions, each question, in turn, having exactly five answer choices, exactly one of which is correct. Every question correctly answered fetches 4 marks and every question incorrectly answered fetches –1 mark. If each student attempted all the questions and the answer choices marked by no two students are the same for all the five questions, then the number of students with a net positive score in the test is at least
Solution
The problem can be solved by considering the possible outcomes for each question.
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For each question, there are 5 possible answers, one of which is correct. This means that for each question, 600 students will answer correctly (since 3000 students / 5 answer choices = 600 students per answer choice). These students will each receive 4 marks for that question.
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The remaining 2400 students will answer incorrectly and will each receive -1 mark for that question.
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Therefore, for each question, the total number of marks awarded will be (600 students * 4 marks) + (2400 students * -1 mark) = 2400 - 2400 = 0 marks.
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Since the total number of marks awarded for each question is 0, the total number of marks awarded for all 5 questions will also be 0.
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This means that the average score for all students is 0.
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However, since scores can be negative (due to incorrect answers), this means that there must be at least one student with a positive score.
Therefore, the number of students with a net positive score in the test is at least 1.
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