Combinatorics

COMBINATORICS

COMBINATORICSFUNDAMENTAL PRINCIPLE OF COUNTING, PERMUTATION, & COMBINATION

RB ASTILLEROAssist. Prof I

RB Astillero1

State the fundamental principle of counting. Give the formula of permutations of n things taken r at a time.Solve permutation problems.Give the formula of combinations of n things taken r at a time.Solve combination problems.

LESSON OBJECTIVES:RB Astillero2

If one event can occur in m ways and a second event can occur in n ways, then both events can occur in m n ways, provided the outcome of the first event does not influence the outcome of the second event. Note: The fundamental principle of counting can be extended to any number of events. When the outcome of one event does not influence the outcome of the other, such events are said to be independent events.

FUNDAMENTAL PRINCIPLE OF COUNTINGRB Astillero3

A restaurant offers a choice of 2 salads, 6 main dishes, 4 side dishes, and 3 desserts. How many different 4-course meals can be selected? Note: Four independent events are involved: selecting a salad, selecting a main dish, selecting a side dish, and selecting a dessert. The first event can occur in 2 ways, the second in 6 ways, the third in 4 ways, and the fourth in 3 ways. Thus, there are 2 6 4 3 = 144 possible meals.

EXAMPLE 1:RB Astillero4

David has 6 books that he wishes to arrange on his desk. How many different arrangements are possible? Note: Six events are involved: selecting a book for the first spot, selecting a book for the second spot, and so on. The outcome of the first event does not influence the outcome of the other events since one book has already been chosen. David has 6 choices for the first spot, 5 choices for the second spot, 4 choices for the third spot, and so on. Thus, there are 6 5 4 3 2 1 = 720 different arrangements for the books.


How many different 7-digit telephone numbers are possible if the first digit cannot be zero and no digit may repeat? Note: Our number system consists of 10 digits, which are {0, 1, 2, 3, 4, 5, 6, 7, 8, and 9}. Since the first digit cannot be a zero, then there are 9 choices for the first digit, 9 choices for the second digit since a zero can be used and no digits may repeat, 8 choices for the third digit, 7 choices for the fourth digit, and so on. Thus, there are 9 9 8 7 6 5 4 =544,320 possible telephone numbers.


When the number of events is few, we can easily determine the possible outcomes by listing all them in a tree diagram.ILLUSTRATION:Consider the 3 letter words that can be made from the letters WORD if no letter is repeated. These can be listed by means of a tree diagram.THE TREE DIAGRAMRB Astillero7

There are:4 ways of choosing the 1st letter3 ways of choosing the 2nd letter2 ways of choosing the 3rd letternumber of words = 4 x 3 x 2 = 24

RB Astillero8

RB Astillero9

How many different arrangements, each consisting of four different letters, can be formed from the letters of the word PERSONAL if each arrangement is to begin and end with a vowel?Ans. 180N = 3652 = 180

In how many different orders can 4 men and 4 women be seated in a row so that no two persons of the same sex will be together?Ans. 1152N = 2(44332211) = 1152

In how many different orders can 7 books be arranged on a shelf if a certain 3-volume work is not to be separated?Ans. 720N= 5(321)(4321) = 720

EXERCISES:RB Astillero10

4. A shelf contains 5 books in red binding, 4 books in blue, and 3 in green. In how many different orders can they be arranged if all books of the same color must be kept together?Ans. 103, 6805. How many different 5-place even numbers can be formed with the digits 1, 2, 3, 4, 5, 6 if repetitions are allowed?Ans. 38886. How many different numbers greater than 400 can be formed with the digits 1, 2, 3, 4, 5 if repetitions are not allowed?Ans. 2647. How many different numbers less than 400 can be formed with the digits 1, 2, 3, 4, 5 if repetitions are not allowed?Ans. 618. A bus has seven vacant seats. If 3 additional passengers enter the bus, in how many different ways can they be seated?Ans. 2109. How many different 3-place numbers can be formed by using 2 odd digits and 1 even digit if zero is excluded but repetitions of digits is allowed?Ans. 300RB Astillero11

10. How many different 3-place numbers can be formed by using 2 odd digits and 1 even digit, it being permissible to use the digit zero and repetition of digits is allowed?Ans. 350

11. If a multiple-choice test consists of 5 questions each with 4 possible answers of which only 1 is correct, in how many ways can a student encircle one answer to each question and get all the answers wrong?Ans. 243

12. How many 3-digit numbers greater than 330 can be formed from the digits 0, 1, 2, 3, 4, 5 if each digit can be used once?Ans. 48RB Astillero12

PERMUTATIONSOther symbol for permutations of n things taken r at a time is nPr.RB Astillero13

Suppose that 7 people enter a swim meet. Assuming that there are no ties, in how many ways could the gold, silver, and bronze medals be awarded? Note: Since awards are to be made, order is important. Thus, this would be treated as a permutation.


Therefore, there are 210 ways that the medals could be awarded.RB Astillero15

In how many ways can 6 students be seated in a row of 6 desks?Note: This is a permutation because order is important. That is, a student may wish to sit in a particular seat or behind/in front of a particular student.EXAMPLE 2:RB Astillero16

There are 720 ways the students could be seated. RB Astillero17

THEOREMS ON PERMUTATIONRB Astillero18

1. In how many different ways can 5 students make a selection of 1 book each from a shelf containing 15 books?Ans. 360, 360N = 15P5 = 360, 3602. In how many different ways can the letters of the word E Q U A T I O N be arranged without changing the place of the consonants?Ans. 720N = (5!)(3!) = 7203. In how many different orders can 5 persons be seated about a round table?Ans. 24N=(5 1)! = 4! = 24

EXERCISES:RB Astillero19

4. In how many different orders can 5 keys be arranged on a ring? Ans. 125. How many distinct permutations can be formed from the letters of the word BANANA? Ans. 606. How many different signals, each consisting of 6 flags hung in a vertical line, can be formed from 4 identical red flags and 2 identical blue flags? Ans. 157. In how many ways can 3 Americans, 4 Frenchmen, 4 Danes and 2 Italians be seated in a row so that those of the same nationality sit together? Ans. 165, 8888. Solve the preceding problem if they sit at a round table. Ans. 41, 4729. Find n if (a) P(n, 2) = 72 and (b) P(n, 4) = 42P(n, 2). Ans. (a) 9, (b) 9RB Astillero20

COMBINATION

RB Astillero21

How many different committees of 3 people can be chosen to work on a special project from a group of 7 people?

Note: Since the order in which the members of the committee are chosen does not affect the result, this is a combination.


There are 35 ways to select the project group.RB Astillero23

In how many different ways can a group of 3 people out of a group of 7 people be chosen to work on a project if it has already been decided that a certain person must work on the project? Note: This is still a combination but the problem has been reduced to selecting 2 more people from the remaining 6 people.


There are 15 ways to select the project group. RB Astillero25

THEOREMS ON COMBINATIONRB Astillero26

1. How many possible combinations can be drawn from a 6/55 lotto draw?Ans. 28, 989, 675N = 55C6 = 28, 989, 6752. How many committees of 5 can be formed from 8 sophomores and 4 freshmen if each committee is to consist of three sophomores and two freshmen?Ans. 336N = 8C34C2 = 3363. In how many ways can a committee consisting of 3 men and 2 women be chosen from 7 men and 5 women? Ans. 350N = 7C35C2 = 350EXERCISES:RB Astillero27

4. A delegation of 4 students is selected each year from a college to attend the National Student Association annual meeting. (a) In how many ways can the delegation be chosen if there are 12 eligible students? (b) In how many ways if two of the eligible students will not attend the meeting together? (c) In how many ways if two of the eligible students are married and will only attend the meeting together? Ans. (a) 495 (b) 450 (c) 255

5. A student is to answer 8 out of 10 questions on an exam. (a) How many choices has he? (b) How many if he must answer the first 3 questions? (c) How many if he must answer at least 4 of the first 5 questions? Ans. (a) 45 (b) 21 (c) 356. In how many ways can a teacher choose one or more students from six eligible students? Ans. 63RB Astillero28

7. In how many ways can 7 toys be divided among 3 children if the youngest gets 3 toys and each of the others gets 2? Ans. 2108. There are 12 students in a class. In how many ways can the 12 students take 3 different tests if 4 students are to take each test? Ans. 34, 6509. In how many different ways can 6 objects be separated into two equal groups? Ans. 1010. From 8 different pairs of gloves, in how many ways can a right-hand and a left-hand glove be selected without taking a pair?Ans. 5611. In how many ways can a selection of 3 letters be made from the letters a, b, c, d, e, f, g, if repetitions are allowed?Ans. 84 RB Astillero29

END OF THE LESSON

DO THE ASSIGNMENT.RB Astillero30

Combinatorics

Documents

Transcript of Combinatorics