Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 1 12 Counting Just How Many Are There?

Slides:



Advertisements
Similar presentations
Counting and Gambling Section 4Objectives: Understand how to use the fundamental counting principle to analyze counting problems that occur in gambling.
Advertisements

DM. 13. A method for counting outcomes of multi-stage processes If you want to perform a series of tasks and the first task can be done in (a) ways, the.
Chapter 13 sec. 3.  Def.  Is an ordering of distinct objects in a straight line. If we select r different objects from a set of n objects and arrange.
Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 1 12 Counting Just How Many Are There?
How many possible outcomes can you make with the accessories?
1 Learning Objectives for Section 7.4 Permutations and Combinations After today’s lesson you should be able to set up and compute factorials. apply and.
Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.
Permutations and Combinations AII Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish.
Probability Using Permutations and Combinations
Thinking Mathematically
If you have a small bag of skittles and you separate then, you have 5 blue, 6 red, 3 purple, and 5 yellow. What is the probability that you would pick.
Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 13.1, Slide 1 13 Probability What Are the Chances?
COUNTING PRINCIPALS, PERMUTATIONS, AND COMBINATIONS.
College Algebra Fifth Edition
Chapter 7 Logic, Sets, and Counting Section 4 Permutations and Combinations.
Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 13.1, Slide 1 13 Probability What Are the Chances?
Chapter 8 Counting Principles: Further Probability Topics Section 8.2 Combinations.
© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 11 Counting Methods and Probability Theory.
7 Further Topics in Algebra © 2008 Pearson Addison-Wesley. All rights reserved Sections 7.4–7.7.
Permutations and Combinations
Math Duels Competition of Scholars. Rules  The class must be split into 2 groups.  Each group must select a team captain and they must do Rock-Paper-Scissors.
Permutations and Combinations. Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish permutations.
1 Combinations. 2 Permutations-Recall from yesterday The number of possible arrangements (order matters) of a specific size from a group of objects is.
Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 1 12 Counting Just How Many Are There?
Section 10-3 Using Permutations and Combinations.
© The McGraw-Hill Companies, Inc., Chapter 4 Counting Techniques.
Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Chapter 7 Logic, Sets, and Counting Section 4 Permutations and Combinations.
Barnett/Ziegler/Byleen Finite Mathematics 11e1 Learning Objectives for Section 7.4 Permutations and Combinations The student will be able to set up and.
Permutations and Combinations
Permutations and Combinations. Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish permutations.
Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 1 12 Counting Just How Many Are There?
Lesson 0.4 (Counting Techniques)
37. Permutations and Combinations. Fundamental Counting Principle Fundamental Counting Principle states that if an event has m possible outcomes and another.
Classwork: wks Homework (day 7): Wks (1-17)/Quiz next block Bring in items to work on Project in class!
Permutations and Combinations
COUNTING Permutations and Combinations. 2Barnett/Ziegler/Byleen College Mathematics 12e Learning Objectives for Permutations and Combinations  The student.
Permutations and Combinations AII Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish.
Copyright © Cengage Learning. All rights reserved. Probability and Statistics.
Permutations and Combinations.  Permutation- An arrangement of objects in which order is important.  Linear Permutation- The arrangement of objects.
Permutations and Combinations. Fundamental Counting Principle Fundamental Counting Principle states that if an event has m possible outcomes and another.
© 2010 Pearson Prentice Hall. All rights reserved. 1 §11.5, Probability with the Fundamental Counting Principle, Permutations, and Combinations.
MAT 110 Workshop Created by Michael Brown, Haden McDonald & Myra Bentley for use by the Center for Academic Support.
Permutations and Combinations
Chapter 10 Counting Methods.
Permutations and Combinations
Unit 4 – Combinatorics and Probability Section 4
8.3 Counting Apply the fundamental counting principle
Chapter 7 Logic, Sets, and Counting
Permutation – The number of ways to ARRANGE ‘n’ items ‘r’ at
Permutations and Combinations
Lesson 11-1 Permutations and Combinations
Permutations and Combinations
Counting Methods and Probability Theory
Permutations and Combinations
Chapter 10 Counting Methods 2012 Pearson Education, Inc.
MATH 2311 Section 2.1.
Chapter 10 Counting Methods.
Counting Methods and Probability Theory
Bellwork Practice Packet 10.3 B side #3.
Counting Methods and Probability Theory
Counting Methods and Probability Theory
Permutations and Combinations
Standard DA-5.2 Objective: Apply permutations and combinations to find the number of possibilities of an outcome.
Permutations and Combinations
Permutations and Combinations
MATH 2311 Section 2.1.
Presentation transcript:

Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 1 12 Counting Just How Many Are There?

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 2 Permutations and Combinations 12.3 Calculate the number of permutations of n objects taken r at a time. Use factorial notation to represent the number of permutations of a set of objects.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 3 Permutations and Combinations 12.3 Calculate the number of combinations of n objects taken r at a time. Apply the theory of permutations and combinations to solve counting problems.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 4 Permutations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 5 Example: How many permutations are there of the letters a, b, c, and d ? Write the answer using P( n, r ) notation. Permutations (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 6 Example: How many permutations are there of the letters a, b, c, and d ? Write the answer using P( n, r ) notation. Solution: We write the letters a, b, c, and d in a line without repetition, so abcd and bcad are two such permutations. Permutations (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 7 The slot diagram indicates there are 4 × 3 × 2 × 1 = 24 possibilities. This can be written more succinctly as Permutations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 8 Example: How many permutations are there of the letters a, b, c, d, e, f, and g if we take the letters three at a time? Write the answer using P(n, r) notation. Permutations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 9 Example: How many permutations are there of the letters a, b, c, d, e, f, and g if we take the letters three at a time? Write the answer using P(n, r) notation. Permutations Solution: The slot diagram indicates there are 7 × 6 × 5 = 210 possibilities. This can be written in permutation notation as shown.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 10 Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 11 Example: Compute (8 – 3)!. Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 12 Example: Compute (8 – 3)!. Solution: We work inside parentheses first. Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 13 Example: Compute. Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 14 Example: Compute. Solution: Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 15 Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 16 Example: The 12-person theater group wishes to select one person to direct a play, a second to supervise the music, and a third to handle publicity, tickets, and other administrative details. In how many ways can the group fill these positions? Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 17 Example: The 12-person theater group wishes to select one person to direct a play, a second to supervise the music, and a third to handle publicity, tickets, and other administrative details. In how many ways can the group fill these positions? Solution: This is a permutation of selecting 3 people from 12. Factorial Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 18 Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 19 Example: How many three-element sets can be chosen from a set of five objects? Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 20 Example: How many three-element sets can be chosen from a set of five objects? Solution: Order is not important, so it is clear that this is a combination problem. Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 21 Example: How many four-person committees can be formed from a set of 10 people? Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 22 Example: How many four-person committees can be formed from a set of 10 people? Solution: Order is not important, so it is clear that this is a combination problem. Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 23 Example: A syndicate has $15 million to spend on tickets for a lottery. Tickets cost $1 and contain a combination of six numbers from 1 to 44. Does the syndicate have enough money to buy enough tickets to be guaranteed a winner? Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 24 Example: A syndicate has $15 million to spend on tickets for a lottery. Tickets cost $1 and contain a combination of six numbers from 1 to 44. Does the syndicate have enough money to buy enough tickets to be guaranteed a winner? Solution : The combination of 6 numbers from the 44 possible gives the number of different tickets. That is, C(44, 6) = 7,059,052. Thus, the syndicate has more than enough money to buy enough tickets to be guaranteed a winner. Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 25 Example: In the game of poker, five cards are drawn from a standard 52-card deck. How many different poker hands are possible? Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 26 Example: In the game of poker, five cards are drawn from a standard 52-card deck. How many different poker hands are possible? Solution : Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 27 Example: In the game of bridge, a hand consists of 13 cards drawn from a standard 52- card deck. How many different bridge hands are there? Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 28 Example: In the game of bridge, a hand consists of 13 cards drawn from a standard 52- card deck. How many different bridge hands are there? Solution : Combinations

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 29 Example: Two men and two women from a firm will attend a conference. If the firm has ten men and nine women, in how many different ways can the conference attendees be selected? Combining Counting Methods (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 30 Example: Two men and two women from a firm will attend a conference. If the firm has ten men and nine women, in how many different ways can the conference attendees be selected? Solution : The answer is not C(19, 4) since this includes options like four men and no woman being sent to the conference. Combining Counting Methods (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 31 Stage 1: Select the two women from the nine available. Stage 2: Select the two men from the ten available. Thus, choosing the women and then choosing the men can be done in ways. Combining Counting Methods

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 32 Example: A 16-member consortium wishes to choose a committee consisting of a president, a vice president, and a three-member executive board. In how many different ways can this committee be formed? Combining Counting Methods (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 33 Example: A 16-member consortium wishes to choose a committee consisting of a president, a vice president, and a three-member executive board. In how many different ways can this committee be formed? Solution : We will count this in two stages: (a) choosing the president and vice president from the consortium, (b) choosing an executive board from the remaining members. Combining Counting Methods (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 34 Stage 1: Choose the president and vice president. This can be done in P(16, 2) ways. Stage 2: Select the executive board. This can be done in C(14, 3) ways. Total: Combining Counting Methods

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 35 Combining Counting Methods

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 36 Combining Counting Methods For example, consider the set {1, 2, 3, 4} and the 4 th row of Pascal’s triangle:

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 37 Combining Counting Methods For example, consider the 4 th row of Pascal’s triangle: C(4, 0) = 1 C(4, 1) = 4 C(4, 2) = 6 C(4, 3) = 4 C(4, 4) = 1

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 38 Example: Assume that a pharmaceutical company has developed five antibiotics and four immune system stimulators. In how many ways can we choose a treatment program consisting of three antibiotics and two immune system stimulators to treat a disease? Use Pascal’s triangle to speed your computations. Combining Counting Methods (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 39 Example: Assume that a pharmaceutical company has developed five antibiotics and four immune system stimulators. In how many ways can we choose a treatment program consisting of three antibiotics and two immune system stimulators to treat a disease? Use Pascal’s triangle to speed your computations. Solution : We will count this in two stages: (a) choosing the antibiotics, (b) choosing the immune system simulators. Combining Counting Methods (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 40 Stage 1: Choosing 3 antibiotics from 5 can be done in C(5, 3) ways. Stage 2: Choosing 2 immune system simulators from 4 can be done in C(4, 2) ways. Total: C(5, 3) × C(4, 2) = 10 × 6 = 60 ways. Combining Counting Methods