Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.

Slides:



Advertisements
Similar presentations
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.
Advertisements

Permutations and Combinations
Math 221 Integrated Learning System Week 2, Lecture 1
5.4 Counting Methods Objectives: By the end of this section, I will be able to… 1) Apply the Multiplication Rule for Counting to solve certain counting.
Chapter 2 Section 2.4 Permutations and Combinations.
Chapter 7 - Part Two Counting Techniques Wednesday, March 18, 2009.
Copyright © Ed2Net Learning Inc.1. 2 Warm Up 1.List all combinations of Roshanda, Shelli, Toshi, and Hector, taken three at a time 2. In how many ways.
Chapter 8 Counting Techniques PASCAL’S TRIANGLE AND THE BINOMIAL THEOREM.
Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF.
P ERMUTATIONS AND C OMBINATIONS Homework: Permutation and Combinations WS.
Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 8 - Slide 1 P-8 Probability The Counting Principle and Permutations.
The Fundamental Counting Principle and Permutations
Chapter 12 Section 8 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND.
Chapter 12 Section 7 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND.
Chapter 11: Counting Methods
Warm Up 1/31/11 1. If you were to throw a dart at the purple area, what would be the probability of hitting it? I I 5.
Chapter 12 Section 7 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1.
Chapter 10: Counting Methods
Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.
10.3 – Using Permutations and Combinations Permutation: The number of ways in which a subset of objects can be selected from a given set of objects, where.
Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.
Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Section 10-3 Using Permutations and Combinations.
© The McGraw-Hill Companies, Inc., Chapter 4 Counting Techniques.
Probability Permutations and Combinations.  Permutations are known as any arrangement of distinct objects in a particular _________. Permutations order.
Statistics 1: Elementary Statistics Section 4-7. Probability Chapter 3 –Section 2: Fundamentals –Section 3: Addition Rule –Section 4: Multiplication Rule.
Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.
Section 3.4 Additional Topics in Probability and Counting © 2012 Pearson Education, Inc. All rights reserved. 1 of 88.
SECTION 10-2 Using the Fundamental Counting Principle Slide
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.8 The Counting Principle and Permutations.
3.4 Counting Principles I.The Fundamental Counting Principle: if one event can occur m ways and a second event can occur n ways, the number of ways the.
Permutations – Page 1CSCI 1900 – Discrete Structures CSCI 1900 Discrete Structures Permutations Reading: Kolman, Section 3.1.
Copyright © 2007 Pearson Education, Inc. Slide 8-1.
Learning Objectives for Section 7.4 Permutations and Combinations
Ch Counting Principles. Example 1  Eight pieces of paper are numbered from 1-8 and placed in a box. One piece of paper is drawn from the box, its.
Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.
PERMUTATIONS AND COMBINATIONS BOTH PERMUTATIONS AND COMBINATIONS USE A COUNTING METHOD CALLED FACTORIAL.
Aim: What is the counting rule? Exam Tomorrow. Three Rules Sometimes we need to know all possible outcomes for a sequence of events – We use three rules.
Counting Principles Multiplication rule Permutations Combinations.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Probability 3.
I CAN: Use Permutations and Combinations
MATH 2311 Section 2.1. Counting Techniques Combinatorics is the study of the number of ways a set of objects can be arranged, combined, or chosen; or.
8.6 Counting Principles. Listing Possibilities: Ex 1 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from.
© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 11 Counting Methods and Probability Theory.
Permutations Counting where order matters If you have two tasks T 1 and T 2 that are performed in sequence. T 1 can be performed in n ways. T 2 can be.
Discrete Math Section 15.3 Solve problems using permutations and combinations Read page Combinations and permutations.
Section The Pigeonhole Principle If a flock of 20 pigeons roosts in a set of 19 pigeonholes, one of the pigeonholes must have more than 1 pigeon.
Chapter 10 Counting Methods.
MATH 2311 Section 2.1.
Permutations and Combinations
Counting Methods and Probability Theory
Counting, Permutations, & Combinations
Chapter 10: Counting Methods
Warm Up Permutations and Combinations Evaluate  4  3  2  1
Combinatorics: Combinations
Section 12.8 The Counting Principle and Permutations
Permutations and Combinations
Combinations.
MATH 2311 Section 2.1.
Counting Methods and Probability Theory
Chapter 10 Counting Methods.
Using Permutations and Combinations
Bellwork Practice Packet 10.3 B side #3.
Chapter 10 Counting Methods 2012 Pearson Education, Inc.
Chapter 11: Further Topics in Algebra
Permutations and Combinations
Using Permutations and Combinations
10.3 – Using Permutations and Combinations
MATH 2311 Section 2.1.
Presentation transcript:

Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 11: Counting Methods 11.1 Counting by Systematic Listing 11.2 Using the Fundamental Counting Principle 11.3 Using Permutations and Combinations 11.4 Using Pascal’s Triangle 11.5 Counting Problems Involving “Not” and “Or”

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 11-3 Using Permutations and Combinations

© 2008 Pearson Addison-Wesley. All rights reserved Using Permutations and Combinations Permutations Combinations Guidelines on Which Method to Use

© 2008 Pearson Addison-Wesley. All rights reserved Permutations In the context of counting problems, arrangements are often called permutations; the number of permutations of n things taken r at a time is denoted n P r. Applying the fundamental counting principle to arrangements of this type gives n P r = n(n – 1)(n – 2)…[n – (r – 1)].

© 2008 Pearson Addison-Wesley. All rights reserved Factorial Formula for Permutations The number of permutations, or arrangements, of n distinct things taken r at a time, where r n, can be calculated as

© 2008 Pearson Addison-Wesley. All rights reserved Example: Permutations Evaluate each permutation. a) 5 P 3 b) 6 P 6 Solution

© 2008 Pearson Addison-Wesley. All rights reserved Example: IDs How many ways can you select two letters followed by three digits for an ID if repeats are not allowed? Solution There are two parts: 1. Determine the set of two letters. 2. Determine the set of three digits. Part 1 Part 2

© 2008 Pearson Addison-Wesley. All rights reserved Example: Building Numbers From a Set of Digits How many four-digit numbers can be written using the numbers from the set {1, 3, 5, 7, 9} if repetitions are not allowed? Solution Repetitions are not allowed and order is important, so we use permutations:

© 2008 Pearson Addison-Wesley. All rights reserved Combinations In the context of counting problems, subsets, where order of elements makes no difference, are often called combinations; the number of combinations of n things taken r at a time is denoted n C r.

© 2008 Pearson Addison-Wesley. All rights reserved Factorial Formula for Combinations The number of combinations, or subsets, of n distinct things taken r at a time, where r n, can be calculated as Note:

© 2008 Pearson Addison-Wesley. All rights reserved Example: Combinations Evaluate each combination. a) 5 C 3 b) 6 C 6 Solution

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding the Number of Subsets Find the number of different subsets of size 3 in the set {m, a, t, h, r, o, c, k, s}. Solution A subset of size 3 must have 3 distinct elements, so repetitions are not allowed. Order is not important.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding the Number of Poker Hands A common form of poker involves hands (sets) of five cards each, dealt from a deck consisting of 52 different cards. How many different 5-card hands are possible? Solution Repetitions are not allowed and order is not important.

© 2008 Pearson Addison-Wesley. All rights reserved Guidelines on Which Method to Use PermutationsCombinations Number of ways of selecting r items out of n items Repetitions are not allowed Order is important.Order is not important. Arrangements of n items taken r at a time Subsets of n items taken r at a time n P r = n!/(n – r)! n C r = n!/[ r!(n – r)!] Clue words: arrangement, schedule, order Clue words: group, sample, selection