# Discrete Structure lecture_29 The pigeon Hole Principle

Apr 28th, 2015
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DISCREAT STRUCTURELECTURE # 29THE PIGEONHOLEPRINCIPLEKHAWAJA TARIQ MAHMOODLecture # 28THE INCLUSION-EXCLUSION PRINCIPLETHE PIGEONHOLE PRINCIPLE:A function from a set of k + 1 or more elements to a set of k elements musthave at least two elements in the domain that have the same image in the co-domain.If k + 1 or more pigeons fly into k pigeonholes then at least one pigeonhole must contain two ormore pigeons.EXAMPLE:1. Among any group of 367 people, there must be at least two with the same birthday,because there are only 366 possible birthdays.2. In any set of 27 English words, there must be at least two that begin with the same letter,since there are 26 letters in the English alphabet.EXAMPLE:What is the minimum number of students in a class to be sure that two of them areborn in the same month?SOLUTION:There are 12 (= n) months in a year. The pigeonhole principle shows that amongany 13 (= n + 1) or more students there must be at least two students who are born in the samemonth.EXAMPLE:Given any set of seven integers, must there be two that have the same remainderwhen divided by 6?SOLUTION:The set of possible remainders that can be obtained when an integer is divided bysix is {0, 1, 2, 3, 4, 5}. This set has 6 elements. Thus by the pigeonhole principle if 7 = 6 + 1integers are each divided by six, then at least two of them must have the same remainder.EXAMPLE:How many integers from 1 through 100 must you pick in order to be sure of

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