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vb.net code 128 font Other Methods of Proof in Java
25 Other Methods of Proof UPC A Creator In Java Using Barcode maker for Java Control to generate, create UPCA image in Java applications. UPC A Recognizer In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. We give here a number of examples that illustrate proof techniques other than direct proof, proof by contradiction, and induction Barcode Generation In Java Using Barcode printer for Java Control to generate, create barcode image in Java applications. Bar Code Reader In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. 251 COUNTING ARGUMENTS
Creating UPC Symbol In C# Using Barcode maker for VS .NET Control to generate, create GS1  12 image in VS .NET applications. UPC A Creator In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create UPCA Supplement 5 image in ASP.NET applications. EXAMPLE 29 Show that if there are 23 people in a room then the odds are better than even that two of them have the same birthday Solution: The best strategy is to calculate the odds that no two people have the same birthday, and then to take complements Let us label the people p1 , p2 , , p23 Then, assuming that none of the p j have the same birthday, we see that p1 can have his birthday on any of the 365 days in the year, p2 can then have his birthday on any of the remaining 364 days, p3 can have his birthday on any of the remaining 363 days, and so forth So the number of different ways that these 23 people can all have different birthdays is 365 364 363 345 344 343 UPC A Maker In .NET Using Barcode creation for Visual Studio .NET Control to generate, create UCC  12 image in .NET framework applications. Drawing UPCA In Visual Basic .NET Using Barcode printer for VS .NET Control to generate, create UPC A image in VS .NET applications. Discrete Mathematics Demystified
GS1  13 Creator In Java Using Barcode creator for Java Control to generate, create UPC  13 image in Java applications. Code 128C Creation In Java Using Barcode encoder for Java Control to generate, create Code 128C image in Java applications. On the other hand, the number of ways that birthdays could be distributed (with no restrictions) among 23 people is 365 365 365 365 = 36523 GS1 DataBar Expanded Drawer In Java Using Barcode printer for Java Control to generate, create GS1 RSS image in Java applications. Data Matrix 2d Barcode Creation In Java Using Barcode printer for Java Control to generate, create Data Matrix ECC200 image in Java applications. 23 times
USD  8 Generation In Java Using Barcode printer for Java Control to generate, create USD8 image in Java applications. Data Matrix 2d Barcode Recognizer In VB.NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. Thus the probability that these 23 people all have different birthdays is p= 365 364 363 343 36523 Generating Barcode In ObjectiveC Using Barcode encoder for iPad Control to generate, create barcode image in iPad applications. Decoding Bar Code In Java Using Barcode Control SDK for Eclipse BIRT Control to generate, create, read, scan barcode image in Eclipse BIRT applications. A quick calculation with a calculator shows that p 04927 < 05 That is the desired result
Generating UCC128 In .NET Framework Using Barcode creator for ASP.NET Control to generate, create GS1128 image in ASP.NET applications. Generating EAN128 In None Using Barcode generation for Word Control to generate, create UCC128 image in Office Word applications. EXAMPLE 210 Jill is dealt a poker hand of 5 cards from a standard deck of 52 What is the probability that she holds four of a kind Solution: If the hand holds 4 aces, then the fth card is any one of the other 48 cards If the hand holds 4 kings, then the fth card is any one of the other 48 cards And so forth So there are a total of 13 48 = 624 possible hands with four of a kind The total number of possible 5card hands is 52 = 2598960 5 Therefore the probability of holding 4 of a kind is p= 624 = 000024 2598960 EAN / UCC  14 Generation In None Using Barcode generation for Online Control to generate, create UCC.EAN  128 image in Online applications. Making Bar Code In Java Using Barcode generation for BIRT Control to generate, create bar code image in BIRT applications. 252 OTHER ARGUMENTS
EXAMPLE 211 Let us show that there exist irrational numbers a and b such that a b is rational
Methods of Mathematical Proof
Solution: Let = 2 and = 2 If is rational then we are done, using a = and b = If is irrational, then observe that 2 = [ = 2 = [ 2]2 = 2 Thus, with a = and b = that a b = 2 is rational
2 we have found two irrational numbers a, b such
EXAMPLE 212 Show that if there are six people in a room then either three of them know each other or three of them do not know each other (Here three people know each other if each of the three pairs has met Three people do not know each other if each of the three pairs has not met) Solution: The tedious way to do this problem is to write out all possible acquaintance assignments for 6 people We now describe a more ef cient, and more satisfying, strategy Call one of the people Bob There are ve others Either Bob knows three of them, or he does not know three of them Say that Bob knows three of the others If any two of those three are acquainted, then those two and Bob form a mutually acquainted threesome If no two of those three know each other, then those three are a mutually unacquainted threesome Now suppose that Bob does not know three of the others If any two of those three are unacquainted, then those two and Bob form an unacquainted threesome If all pairs among the three are instead acquainted, then those three form a mutually acquainted threesome We have covered all possibilities, and in every instance come up either with a mutually acquainted threesome or a mutually unacquainted threesome That ends the proof It may be worth knowing that ve people is insuf cient to guarantee either a mutually acquainted threesome or a mutually unacquainted threesome We leave it to the reader to provide a suitable counterexample It is quite dif cult to determine the minimal number of people to solve the problem when threesome is replaced by foursome When foursome is replaced by ve people, the problem is considered to be grossly intractable This problem is a simple example from the mathematical subject known as Ramsey theory

