The Fundamental Counting Principle

1. Chapter 7 Combinatorics 7.1 The Fundamental Counting Principle 7.1.1 MATHPOWERTM 12, WESTERN EDITION

2. Using a Tree Diagram to Determine the Number of Combinations 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 Desserts Combinations Main Courses cheesecake pie soup, pasta, cheesecake pasta steak fish soup, pasta, pie cheesecake pie soup, steak, cheesecake soup soup, steak, pie cheesecake pie soup, fish, cheesecake soup, fish, pie cheesecake pie salad, pasta, cheesecake pasta steak fish salad, pasta, pie cheesecake pie salad, steak, cheesecake salad salad, steak, pie cheesecake pie salad, fish, cheesecake 7.1.2 salad, fish, pie

3. Using a Tree Diagram Vs. The Fundamental Counting Principle From the tree diagram, we can see that Stan can choose from 12 different meals. 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. The Fundamental Counting Principle: If a task is made up of many stages, the total number of possibilities for the task is given by mxnxpx . . . 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. 7.1.3

4. Applying The Fundamental Counting Principle • 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? 6 4 _____ x ______ x ______ 4 = 96 ways Skirts Blouses Sweaters Colleen can choose from 96 different outfits. • The final score in a soccer game is 5 to 4 for Team A. • How many different half-time scores are possible? There are 30 different possible half-time scores. _____ x ______ 6 5 Team A (0 - 5) Team B (0 - 4) 7.1.4

5. Applying The Fundamental Counting Principle 3. How many four-digit numerals are are there with no repeated digits? _____ x ______ x ______ x _____ 9 9 8 7 = 4536 1st 2nd 3rd 4th can be zero, but can’t be the same as the first digit The number of four-digits numerals with no repeated digits is 4536. can’t be zero two digits have been used three digits have been used 4. How many odd four-digit numerals have no repeated digits? _____ x ______ x ______ x _____ 8 8 7 5 = 2240 1st 2nd 3rd 4th The number of odd four-digit numerals with no repeated digits is 2240. must be odd: 1, 3, 5, 7, or 9 can’t be zero or the same as the last digit three digits have been used two digits have been used 7.1.5

6. Applying the Fundamental Counting Principle 5. How many even four-digit numerals have no repeated digits? There are two cases which must be considered when solving this problem: 1. zero not the last digit 2. zero as the last digit 1. _____ x ______ x ______ x _____ 8 8 7 4 = 1792 1st 2nd 3rd 4th can’t be zero or the same as the last digit two digits have been used three digits have been used must be even but not zero: 2, 4, 6, or 8 The number of even four-digit numerals with no repeated digits is 1792 + 504 = 2296. AND 9 1 8 7 2. _____ x ______ x ______ x _____ = 504 1st 2nd 3rd 4th can’t be zero two digits have been used three digits have been used must be zero 7.1.6

7. Applying the Fundamental Counting Principle 6. How many four-letter arrangements are possible? _____ x ______ x ______ x _____ 26 26 26 26 = 456 976 The number of four-letter arrangements is 456 976. 1st 2nd 3rd 4th use any of the 26 letters: A - Z repetition is allowed repetition is allowed repetition is allowed • 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 ____ 4 4 4 4 4 4 4 4 4 4 1 2 3 4 5 6 7 8 9 10 You can complete the test 410 or 1 048 576 ways. 7.1.7

8. Applying the Fundamental Counting Principle 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 ____ 4 3 3 There are 36 three-letter arrangements. 1st2nd3rd 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 consonant: C, R, T, or N 9. How many three-digit numerals less than 500 can be formed using the digits 1, 3, 4, 6, 8, and 9? ____ x ____ x ____ 3 6 6 There are 108 three-digit numbers. 1st2nd3rd 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 7.1.8

9. Assignment Suggested Questions Pages 336 and 337 4-13, 14 a 7.1.6