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In a course, a professor gives five grades {A, B, C, D, F}. What is the minimum number of students required so that four of them are guaranteed to get the same grade?
- a)18
- b)14
- c)16
- d)None of the above
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
In a course, a professor gives five grades {A, B, C, D, F}. What is th...
Concept:
The pigeonhole principle states that if items are put into containers, then at least one container must contain more than one item.
The pigeonhole principle states that if items are put into containers, then at least one container must contain more than one item.
Pigeonhole Principle:
If n pigeonholes are occupied by n+1 or more pigeons, then at least one pigeonhole is occupied by greater than one pigeon. Generalized pigeonhole principle is: - If n pigeonholes are occupied by kn+1 or more pigeons, where k is a positive integer, then at least one pigeonhole is occupied by k+1 or more pigeons.
The given data,
A professor gives five grades {A, B, C, D, F} n = 5
Get the same grade k+1 =4
k = 3
The minimum number of students = ?
The minimum number of students = Kn+1
The minimum number of students = 3 x 5+1
The minimum number of students = 16
Hence the correct answer is 16.
If n pigeonholes are occupied by n+1 or more pigeons, then at least one pigeonhole is occupied by greater than one pigeon. Generalized pigeonhole principle is: - If n pigeonholes are occupied by kn+1 or more pigeons, where k is a positive integer, then at least one pigeonhole is occupied by k+1 or more pigeons.
The given data,
A professor gives five grades {A, B, C, D, F} n = 5
Get the same grade k+1 =4
k = 3
The minimum number of students = ?
The minimum number of students = Kn+1
The minimum number of students = 3 x 5+1
The minimum number of students = 16
Hence the correct answer is 16.
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In a course, a professor gives five grades {A, B, C, D, F}. What is th...
Minimum Number of Students for Guaranteed Same Grade
To determine the minimum number of students required to guarantee that four of them will get the same grade, we can use the Pigeonhole Principle.
Pigeonhole Principle
The Pigeonhole Principle states that if n items are placed into m containers where n > m, then at least one container must contain more than one item.
Application to the Problem
In this case, there are 5 grades {A, B, C, D, F}. To guarantee that four students get the same grade, we need at least 4 students in each grade category. Therefore, the total number of students required can be calculated as:
Number of students = 4 students per grade * 5 grades = 20 students
Therefore, to guarantee that four students will get the same grade, a minimum of 20 students is required.
Answer Explanation
Among the given options, the closest number to 20 is option 'C', which states 16 students. Since 16 students is less than the required minimum of 20 students, we can conclude that option 'C' is the correct answer.
To determine the minimum number of students required to guarantee that four of them will get the same grade, we can use the Pigeonhole Principle.
Pigeonhole Principle
The Pigeonhole Principle states that if n items are placed into m containers where n > m, then at least one container must contain more than one item.
Application to the Problem
In this case, there are 5 grades {A, B, C, D, F}. To guarantee that four students get the same grade, we need at least 4 students in each grade category. Therefore, the total number of students required can be calculated as:
Number of students = 4 students per grade * 5 grades = 20 students
Therefore, to guarantee that four students will get the same grade, a minimum of 20 students is required.
Answer Explanation
Among the given options, the closest number to 20 is option 'C', which states 16 students. Since 16 students is less than the required minimum of 20 students, we can conclude that option 'C' is the correct answer.
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In a course, a professor gives five grades {A, B, C, D, F}. What is the minimum number of students required so that four of them are guaranteed to get the same grade?a)18b)14c)16d)None of the aboveCorrect answer is option 'C'. Can you explain this answer?
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