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$
2011 AMC 10B Problem 11
The Pigeon Hole Principle
6 out of 10
Mathematics Circle
You have $52$ people in a room. You have to place them in $12$ boxes.
can you say why did i take $12$ boxes?
Because there are $12$ months in year.
One box must have at least $ \frac {52} {12} $
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