Pigeonhole Principle Research Paper

THE PIGEONHOLE PRINCIPLE
ABSTRACT
Pigeonhole principle is a part of combinatorics and has the base of permutations and combinations. It is used to solve a variety of problems most of them ending with unexpected results. Known by a variety of names such as Pigeonhole Principle, Dirichlet Drawer Principle, Shelf Principle and Shoebox Principle [1], it states that if n items are put into n – 1 compartments then at least one of the compartment will have more than one object. It is used widely in computer science, computer programming and in many more fields.
Keywords: Combinatorics, permutations and combinations, pigeonhole principle.
INTRODUCTION
The Pigeonhole Principle is believed to have been discovered by Peter Gustav Lejeune Dirichlet under

If for each i = 1, 2,….., n the ith box contains fewer than qi objects, then the total number of objects in all the boxes does not exceed (q1 – 1) + (q2 – 2) +…..+( qn – 1) = q1 + q2 +…….+ qn – n. Since this number is one less than the number of objects distributed, we conclude that for some i = 1, 2,…, n the ith box contains at least qi objects. THEOREM 3: If the average of n nonnegative integers m1, m2,….,mn is greater than r-1 that is, (m1+m2+⋯+mn)/n > r – 1, then at least one of the integers is greater than or equal to r. EXAMPLES SOCK-PICKING
Assume a drawer contains a mixture of black socks and blue socks, each of which can be worn on either foot, and that you are pulling a number of socks from the drawer without looking. What is the minimum number of pulled socks required to guarantee a pair of the same colour? Using the pigeonhole principle, to have at least one pair of the same colour (m = 2 holes, one per colour) using one pigeonhole per colour, you need to pull only three socks from the drawer (n = 3 items). Either you have three of one colour, or, exclusively, two of one colour and one of the other.

As the ‘holes’, or m, correspond to number of hands shaken, and each person can shake hand with anybody from 0 to n – 1 other people, this creates n – 1 possible holes. This is because either the ‘0’ or the ‘n – 1’ hole must be empty (if one person shakes hands with everybody, it’s not possible to have another person who shakes hands with nobody; likewise, if one person shakes hands with no one there cannot be a person who shakes hands with everybody). This leaves n people to be placed in at most n – 1 non-empty holes, guaranteeing duplication.

THE BIRTHDAY PROBLEM
The birthday problem asks, for a set of n randomly chosen people, what is the probability that some pair of them will have the same birthday? By the pigeonhole principle, if there are 367 people in the room, we know that there is at least one pair who shares the same birthday, as there are only 366 possible birthdays to choose from (including February 29, if present).

THE SOFTBALL