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The **Pigeon Hole Principle** (also known as the *Dirichlet box principle*, *Dirichlet principle* or *box principle*) states that if $ \textbf n+1 $ or more pigeons are placed in $ \textbf n $ holes, then one hole must contain two or more pigeons.

The extended version of this Principle states that if $ \textbf k$ objects are placed in $ \textbf n$ boxes then at least one box must hold at least $ \frac {k} {n} $ objects.

There are $52$ people in a room. what is the largest value of $ \textbf n $ such that the statement "At least $ \textbf n $ people in this room have birthdays falling in the same month" is always true?

$ \textbf {(A)} 2\quad \textbf {(B)} 3\quad \textbf {(C)} 4\quad \textbf {(D)} 5\quad \textbf {(E)} 12$

Source

Competency

Difficulty

Suggested Book

2011 AMC 10B Problem 11

The Pigeon Hole Principle

6 out of 10

Mathematics Circle

First hint

You have $52$ people in a room. You have to place them in $12$ boxes.

Second Hint

can you say why did i take $12$ boxes?

Because there are $12$ months in year.

Final Step

One box must have at least $ \frac {52} {12} $

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