## What is The Pigeon Hole Principle?

TheÂ **Pigeon Hole Principle**Â (also known as theÂ *Dirichlet box principle*,Â *Dirichlet principle*Â orÂ *box principle*) states that if $ \textbf n+1 $Â or more pigeons are placed in $ \textbf n $Â 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$ Â boxes then at least one box must hold at least $ \frac {k} {n} $Â objects.

## Try the problem

There are $52$Â people in a room. what is the largest value ofÂ $ \textbf n $Â such that the statement “At leastÂ $ \textbf n $Â 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$

2011 AMC 10B Problem 11

The Pigeon Hole Principle

6 out of 10

Mathematics Circle

## Knowledge Graph

## Use some hints

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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