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In a certain test, αi, students gave wrong answers to at least i questions, where i = 1,2,3.4...k. No student gave more than k wrong answers the total number of wrong answers given is :
Correct answer is option ''. Can you explain this answer?
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Total Wrong Answers Calculation
In this problem, we need to analyze the situation where students have given wrong answers to a specific number of questions. The key points are:
- Each student, denoted by αi, has given wrong answers such that:
- α1: at least 1 wrong answer
- α2: at least 2 wrong answers
- α3: at least 3 wrong answers
- ...
- αk: at least k wrong answers
Understanding the Constraints
- Each student has provided wrong answers that are cumulative. This means:
- The first student can have 1 or more wrong answers.
- The second student has at least 2 wrong answers, implying they could have more.
- This pattern continues up to the k-th student who has at least k wrong answers.
Total Wrong Answers
- To calculate the total number of wrong answers given by all students, we can use the following logic:
- For each student αi, the minimum number of wrong answers contributed is i.
- Therefore, if we sum these contributions, the total wrong answers (T) can be computed as:
- T = 1 + 2 + 3 + ... + k
Formula for the Sum
- The sum of the first k natural numbers is given by the formula:
- T = k * (k + 1) / 2
Conclusion
- Thus, the total number of wrong answers given by all students is the sum of the series, resulting in T = k * (k + 1) / 2.
- This formula accounts for all students while adhering to the problem's constraints.
In summary, understanding how each student contributes to the total count is crucial in solving the problem effectively.
In this problem, we need to analyze the situation where students have given wrong answers to a specific number of questions. The key points are:
- Each student, denoted by αi, has given wrong answers such that:
- α1: at least 1 wrong answer
- α2: at least 2 wrong answers
- α3: at least 3 wrong answers
- ...
- αk: at least k wrong answers
Understanding the Constraints
- Each student has provided wrong answers that are cumulative. This means:
- The first student can have 1 or more wrong answers.
- The second student has at least 2 wrong answers, implying they could have more.
- This pattern continues up to the k-th student who has at least k wrong answers.
Total Wrong Answers
- To calculate the total number of wrong answers given by all students, we can use the following logic:
- For each student αi, the minimum number of wrong answers contributed is i.
- Therefore, if we sum these contributions, the total wrong answers (T) can be computed as:
- T = 1 + 2 + 3 + ... + k
Formula for the Sum
- The sum of the first k natural numbers is given by the formula:
- T = k * (k + 1) / 2
Conclusion
- Thus, the total number of wrong answers given by all students is the sum of the series, resulting in T = k * (k + 1) / 2.
- This formula accounts for all students while adhering to the problem's constraints.
In summary, understanding how each student contributes to the total count is crucial in solving the problem effectively.
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