Solving a problem using the Pigeonhole principle How do I use the Pigeonhole principle to show that

Eden Solomon

Eden Solomon

Answered question

2022-06-09

Solving a problem using the Pigeonhole principle
How do I use the Pigeonhole principle to show that in a class of 25 students where every student is either a sophomore, freshman or a junior there are at least 9 sophomores or 9 seniors or 9 juniors ?
My solution: Assume by contradiction that there are 8 sophomores, 8 freshman and 8 juniors then 8 + 8 + 8 = 24 25 thus by contradiction there are at least 9 sophomores or 9 seniors or 9 juniors
Is this solution a correct application of the Pigeonhole principle ?

Answer & Explanation

Jerome Page

Jerome Page

Beginner2022-06-10Added 16 answers

Explanation:
Let a, b, c be the three values which can only be whole numbers. Then their mean is 25 / 3 = 8.33333. Since the mean has to lie in between the values (if not all three values are equal and not outside the range OR if all the values are identical the mean is equal to each of them (which cannot be the case here since the mean is not a positive integer), atleast one of the values must be a positive greater than 8.333... or atleast 9.
Arraryeldergox2

Arraryeldergox2

Beginner2022-06-11Added 10 answers

The one nitpick I have is that, if there are not 9 of a given year, there are at most 8 of the given year. That is, 8 is simply the worst case scenario.

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