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An institute has 5 departments and each department has 50 students. If students are picked up randomly from all 5 departments to, form a committee, what should be the minimum number of students in the committee so that atleast one department should have representation of minimum 5 students?
  • a)
    11
  • b)
    15
  • c)
    21
  • d)
    41
  • e)
    None of the above
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
An institute has 5 departments and each department has 50 students. I...
If we choose maximum 4 students from each department, there will be 20 students. So, 21st student will ensure at least one group has 5 representatives.
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Community Answer
An institute has 5 departments and each department has 50 students. I...
To find the minimum number of students in the committee such that at least one department has a representation of minimum 5 students, we can use the concept of the Pigeonhole Principle.

The Pigeonhole Principle states that if there are more pigeons than there are pigeonholes, then at least one pigeonhole must contain more than one pigeon. In this case, the pigeons represent the students and the pigeonholes represent the departments.

Let's analyze the problem step by step:

1. Each department has 50 students, so there are a total of 5*50 = 250 students in the institute.

2. We want to form a committee with the minimum number of students such that at least one department has a representation of minimum 5 students. This means that we need to distribute the students in such a way that no department has less than 5 students.

3. To minimize the number of students in the committee, we can distribute the first 4 students to each department. This ensures that each department has at least 1 student.

4. Now, we need to distribute the remaining students in such a way that no department has less than 5 students. Since each department already has 1 student, we need to distribute the remaining (250-4) = 246 students.

5. To ensure that no department has less than 5 students, we can distribute the remaining students evenly among the departments. This means each department will receive (246/5) = 49.2 students.

6. Since the number of students must be a whole number, we need to round up the fraction. Therefore, each department will receive at least 50 students.

7. Adding the initial 4 students, the minimum number of students required in the committee is 4 + 50 = 54.

However, the question asks for the minimum number of students required in the committee such that at least one department has a representation of minimum 5 students. Since we have already ensured that each department has at least 5 students, we can subtract one student from any one of the departments without violating the condition.

Therefore, the minimum number of students required in the committee is 54 - 1 = 53.

The closest option provided is 21, which is not the correct answer. The correct answer is option 'C' which is 53.
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An institute has 5 departments and each department has 50 students. If students are picked up randomly from all 5 departments to, form a committee, what should be the minimum number of students in the committee so that atleast one department should have representation of minimum 5 students?a) 11b) 15c) 21d) 41e) None of the aboveCorrect answer is option 'C'. Can you explain this answer?
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