Discrete Structures Counting (Ch. 6)

Discrete StructuresCounting (Ch. 6) Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore. mhumayoun@ciitlahore.edu.pk https://sites.google.com/a/ciitlahore.edu.pk/dstruct/ Modified slides of Dr. M. Atif

First Reaction Counting?? What?? Are we first graders??

The Product Rule • Also called the multiplication rule • If there are n1 ways to do task 1, and n2 ways to do task 2: • Then there are n1n2 ways to do both tasks in sequence • This applies when doing the “procedure” is made up of separate tasks. • We must make one choice AND a second choice

Product Rule Example • Sample question • There are 18 math majors and 25 CS majors • How many ways are there to pick one math major and one CS major? • Total is 18 * 25 = 450 possible ways

Product Rule Examples Ex.2: p-386. The chairs of an auditorium are to be labeled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labeled different? Largest Number of chairs = 26 * 100 = 2600 Ex.3: There are 32 microcomputers in a computer center. Each microcomputer has 24 ports. How many different ports to a microcomputer in the center are there. Two tasks – first picking a microcomputer and then picking a post on this microcomputer. Total no. of ways = 32 * 24 = 768 ports

Product Rule Examples Ex.1: p-386. A new company with just two employees, rents a floor of a building with 12 offices. How many ways are there to assign different offices to these two employees? Possible ways of allotment of any office to first employee = 12 Possible ways of allotment of any of the remaining office to second employee = 11 All possible ways of allotment of offices to two employees = 12 * 11 = 132 See Example 4, 5

Product Rule Examples More sample questions… • How many strings of 4 decimal digits… • Do not contain the same digit twice? • We want to chose a digit, then another that is not the same, then another… • First digit: 10 possibilities • Second digit: 9 possibilities (all but first digit) • Third digit: 8 possibilities • Fourth digit: 7 possibilities • Total = 10*9*8*7 = 5040 • End with an even digit? • First three digits have 10 possibilities • Last digit has 5 possibilities • Total = 10*10*10*5 = 5000

More examples • How many different bit strings of length 7 are there? • 27= 128 different bit strings of length seven. • How many different license plates are available if each plate contains a sequence of three upper case letters followed by three digits. • Choices for 3 upper case letter: 26 * 26 * 26 • Choices for 3 digits: 10 * 10 * 10 • Total choices: 263 · 103

More Examples • What is the value of k after the following code, where n1, n2, . . . , nm are +ve integers: n1* n2* ...* nm

The Sum Rule • Also called the addition rule • If there are n1 ways to do task 1, and n2 ways to do task 2 • If these tasks can be done at the same time, then… • Then there are n1+n2 ways to do one of the two tasks • We must make first choice OR a second choice

Sum Rule Examples More sample questions • How many strings of 4 decimal (non-zero)digits… • Have exactly three digits that are 9s? • The string can have: • The non-9 as the first digit • OR the non-9 as the second digit • OR the non-9 as the third digit • OR the non-9 as the fourth digit • Thus, we use the sum rule • For each of those cases, there are 9 possibilities for the non-9 digit (any number other than 9) • Thus, the answer is 9+9+9+9 = 36

Sum Rule Examples • Sample question • There are 18 math majors and 25 CS majors • How many ways are there to pick one math major orone CS major? • Total is 18 + 25 = 43

n1 + n2 + … + nm

Sum Rule Examples Ex.11: Suppose that either a member of the mathematics faculty or a student who is mathematics major is chosen as a representative to a university committee. How many different choices are there for this representative if there are 37 members of the math faculty and 83 math students. All possible ways to pick a representative = 37 + 83 = 120 See Ex. 12 . Exercise Q. 1,3

Misc Examples There are 18 mathematics majors and 325 computer science majors at a college • In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major? 18 . 325 • In how many ways can one representative be picked who is either a mathematics major or a computer science major? 18 + 325

A multiple-choice test contains 10 questions. There are four possible answers for each question. • In how many ways can a student answer the questions on the test if the student answers every question? 4 * 4 * …. * 4 (10 times)= 410 • In how many ways can a student answer the questions on the test if the student can leave answers blank? 510

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each gender. How many different types of this shirt are made? • 12 * ( 3 + 3)

How many strings of three decimal digits • do not contain the same digit three times? • 10*10*10 – 10 = 990 • begin with an odd digit? • 5* 10*10 = 500 • have exactly two digits that are 4s? • 10 + 10 + 10 = 30

A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee? • (1+1+1)50

How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits? • 10^3 * 26^3 + 26^3 * 10^3

How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters? • 26^2*10^4 + 10^2*26^4

6.2 The Pigeonhole Principle • Suppose that there are 9 pigeonholes for a flock of 10 pigeons. Because there are 10 pigeons but only 9 pigeonholes, a least one of these 9 pigeonholes must have at least two pigeons in it. • This illustrates the pigeonhole principle. If there are more pigeons than pigeonholes, then there must be at least one pigeonhole with at least two pigeons in it.

The Pigeonhole Principle • If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.

Can be used to demonstrate possibly unexpected results • Among a group of 367 people (randomly chosen), there must be at least two with the same birthday, because there are only 365 possible birthdays. • In any group of 27 English words (randomly chosen), there must be at least two that begin with the same letter, because there are 26 letters in the English alphabet. • Among a set of 15 or more students, at least 3 are born on the same day of the week.

More examples Question:How many students must be in a class to guarantee that at least two students receive the same score on the final exam, if the exam is graded on a scale from 0 to 100 points? Ans: There are 101 possible scores on the final. The pigeonhole principle shows that among any 102 students there must be at least 2 students with the same score.

The Generalized Pigeonhole Principle • Even more can be said when the number of objects exceeds a multiple of the number of boxes. • Among any set of 21 decimal digits there must be 3 that are the same. • Why? when 21 objects are distributed into 10 boxes, one box must have more than 2 objects.

The Last slide. VIS • How many numbers must be selected from the set {1, 2, 3, 4 , 5, 6} to guarantee that at least one pair of these numbers add up to 7? • Sol: • Total numbers = 6 • Possible ways to add up to 7= (1,6), (2, 5), (3, 4) • We have to select at least 4 numbers, so that at least one pair from above 3 pairs is selected.

Discrete Structures Counting (Ch. 6)