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Pigeon Hole Principle Problem-11 from 2011 AMC 10B

This is a beautiful problem from 2011 AMC 10B-Problem 11.We ma use sequential hints.

What is The Pigeon Hole Principle?

The Pigeon Hole Principle (also known as the Dirichlet box principleDirichlet 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.

Try the problem

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

Use some hints

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 $\frac {52} {12}$