MATHPOWER TM 12, WESTERN EDITION Chapter 7 Combinatorics

Stan is about to order dinner at a restaurant. He has a choice of two appetizers (soup or salad), three main courses (pasta, steak, or fish), and two desserts (cheesecake or pie). How many different meal combinations can Stan choose? Appetizers soup salad Main Courses pasta steak fish pasta steak fish cheesecake pie cheesecake pie cheesecake pie cheesecake pie cheesecake pie cheesecake pie Desserts soup, pasta, cheesecake soup, pasta, pie soup, steak, cheesecake soup, steak, pie soup, fish, cheesecake soup, fish, pie salad, pasta, pie salad, steak, cheesecake salad, steak, pie salad, fish, cheesecake salad, fish, pie salad, pasta, cheesecake Combinations Using a Tree Diagram to Determine the Number of Combinations 7.1.2

Read on to learn about a different and easier method for determining the number of combinations: There are two choices of appetizer, three main course choices and two dessert choices. This will give Stan a total of 2 x 3 x 2 = 12 meal combinations. This method is called the Fundamental Counting Principle. If a task is made up of many stages, the total number of possibilities for the task is given by m x n x p x... where m is the number of choices for the first stage, and n is the number of choices for the second stage, p is the number of choices for the third stage, and so on. The Fundamental Counting Principle: Using a Tree Diagram Vs. The Fundamental Counting Principle From the tree diagram, we can see that Stan can choose from 12 different meals.

1.Colleen has six blouses, four skirts and four sweaters. How many different outfits can she choose from, assuming she wears three different items at once? _____ x ______ x ______ Blouses Skirts Sweaters 64 4 = 96 ways Colleen can choose from 96 different outfits. 2.The final score in a soccer game is 5 to 4 for Team A. How many different half-time scores are possible? _____ x ______ Team A (0 - 5) Team B (0 - 4) 65 There are 30 different possible half-time scores Applying The Fundamental Counting Principle

3. How many four-digit numerals are are there with no repeated digits? _____ x ______ x ______ x _____ 7998 = 4536 The number of four-digits numerals with no repeated digits is Applying The Fundamental Counting Principle 1st 2nd3rd4th can’t be zero can be zero, but can’t be the same as the first digit two digits have been used three digits have been used 4. How many odd four-digit numerals have no repeated digits? _____ x ______ x ______ x _____ 5887 = 2240 The number of odd four-digit numerals with no repeated digits is must be odd: 1, 3, 5, 7, or 9 1st 2nd3rd4th can’t be zero or the same as the last digit two digits have been used three digits have been used

7.1.6 Applying the Fundamental Counting Principle 5. How many even four-digit numerals have no repeated digits? 1. _____ x ______ x ______ x _____ must be even but not zero: 2, 4, 6, or 8 2. _____ x ______ x ______ x _____ must be zero AND = 1792 = 504 The number of even four-digit numerals with no repeated digits is = st2nd3rd4th can’t be zero or the same as the last digit two digits have been used three digits have been used There are two cases which must be considered when solving this problem: 1. zero not the last digit 2. zero as the last digit 1st 2nd 3rd 4th can’t be zero two digits have been used three digits have been used

7.1.7 Applying the Fundamental Counting Principle 6. How many four-letter arrangements are possible? _____ x ______ x ______ x _____ 26 = The number of four-letter arrangements is st 2nd3rd4th use any of the 26 letters: A - Z repetition is allowed 7.You are given a multiple choice test with 10 questions. There are four possible answers to each question. How many ways can you complete the test? ____ x ____ x ____ x ____ x ____ x ____ x ____ x ____ x ____ x ____ You can complete the test 4 10 or ways. repetition is allowed repetition is allowed

8. How many three-letter arrangements can be made from the letters of the word CERTAIN, if no letter can be used more than once and each is made up of a vowel between two consonants? ____ x ____ x ____ How many three-digit numerals less than 500 can be formed using the digits 1, 3, 4, 6, 8, and 9? ____ x ____ x ____ 1st 2nd 3rd 366 There are 36 three-letter arrangements. There are 108 three-digit numbers Applying the Fundamental Counting Principle 1st 2nd 3rd must be a consonant: C, R, T, or N must be a vowel: E, A, or I must be a consonant and can’t be the same as the first letter must be a digit less than 5: 1, 3, or 4 can be any of the 6 digits can be any of the 6 digits

Suggested Questions Pages 336 and , 14 a 7.1.6