9.5 Counting Principles
- 1. 9.5 Counting Principles Chapter 9 Sequences, Probability, and Counting Theory
- 2. Concepts and Objectives ⚫ The objectives for this section are ⚫ Solve counting problems using the Addition Principle. ⚫ Solve counting problems using the Multiplication Principle. ⚫ Solve counting problems using permutations involving n distinct objects. ⚫ Solve counting problems using combinations. ⚫ Find the number of subsets of a given set. ⚫ Solve counting problems using permutations involving n non-distinct objects.
- 3. Counting Principles ⚫ The Addition Principle states that if one event can occur in m1 ways, and a second event (with no common outcomes) can occur in m2 ways, etc., then the first or second or etc. can occur in m1 + m2 + … ways. ⚫ Example: There are 2 vegetarian entrée options and 5 meat entrée options on a dinner menu. What is the total number of entrée options?
- 4. Counting Principles ⚫ The Addition Principle states that if one event can occur in m1 ways, and a second independent event can occur in m2 ways, etc., then the first or second or etc. can occur in m1 + m2 + … ways. ⚫ Example: There are 2 vegetarian entrée options and 5 meat entrée options on a dinner menu. What is the total number of entrée options? Since each of these options is separate, the total number of options is 2 + 5 = 7.
- 5. Counting Principles (cont.) ⚫ The Multiplication Principle states that if there are m1 ways for one to event occur, m2 ways for the second event to occur after the first event, etc., then the events can occur in m1 × m2 × … ⚫ Example: Diana packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit, and decide whether to wear the sweater. Find the total number of possible outfits.
- 6. Counting Principles (cont.) ⚫ Example: Diana packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit, and decide whether to wear the sweater. Find the total number of possible outfits. To find the total number of outfits, multiply the number of skirt options, the number of blouse options, and the number of sweater options: skirt options × blouse options × sweater options 2 × 4 × 2 = 16
- 7. Counting Principles (cont.) ⚫ Example: As a promotion, a restaurant offered a choice of 3 appetizers, 7 main dishes, and 4 desserts for $9.99. How many different 3-course meals are possible?
- 8. Counting Principles (cont.) ⚫ Example: As a promotion, a restaurant offered a choice of 3 appetizers, 7 main dishes, and 4 desserts for $9.99. How many different 3-course meals are possible? Each course is an event. The first event can occur in 3 ways, the second event can occur in 7 ways, and the third event can occur in 4 ways. Therefore, there are 3 7 4 = 84 possible meals
- 9. Counting Principles (cont.) ⚫ Example: A librarian has 5 different books that she wants to arrange in a row. How many different arrangements are possible?
- 10. Counting Principles (cont.) ⚫ Example: A librarian has 5 different books that she wants to arrange in a row. How many different arrangements are possible? Five events are involved: When we select a book for the first spot, that leaves 4 choices for the second spot. Continuing in this fashion gives us 3 choices for the third spot, 2 choices for the fourth spot, and 1 choice for the fifth spot: 5 4 3 2 1 = 120 different arrangements
- 11. How Do I Know Which One? ⚫ When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes?
- 12. How Do I Know Which One? ⚫ When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? ⚫ The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.
- 13. Permutations ⚫ A permutation of n elements taken r at a time is one of the arrangements of r elements from a set of n elements. ⚫ Other ways to write P(n, r) are and nPr . In Desmos, use the function nPr(n,r). If P(n, r) denotes the number of permutations of n elements taken r at a time, with r n, then ( ) ( ) = − ! , ! n P n r n r n r P
- 14. Permutations (cont.) ⚫ Example: Suppose 12 people enter a race. In how many ways could the gold, silver, and bronze medals be awarded?
- 15. Permutations (cont.) ⚫ Example: Suppose 12 people enter a race. In how many ways could the gold, silver, and bronze medals be awarded? ( ) ( ) = − 12! 12,3 12 3 ! P = 12! 9! =10 11 12 =1320 possibilities
- 16. Combinations ⚫ A subset of items selected without regard to order is called a combination. ⚫ Another way to write C(n, r) is nCr . As mentioned last class, in Desmos, we can use nCr(n,r). If C(n, r) denotes the number of combinations of n elements taken r at a time, with r n, then ( ) ( ) = = − ! , ! ! n n C n r r n r r
- 17. Combinations (cont.) ⚫ Example: How many different committees of 5 people can be chosen from a group of 9 people?
- 18. Combinations (cont.) ⚫ Example: How many different committees of 5 people can be chosen from a group of 9 people? ( ) ( ) = = − 9 9! 9,5 5 9 5 !5! C = 9! 4!5! =126 committees
- 19. How Do I Know Which One? Permutations Combinations Number of ways of selecting r items out of n items
- 20. How Do I Know Which One? Permutations Combinations Number of ways of selecting r items out of n items Repetitions are not allowed
- 21. How Do I Know Which One? Permutations Combinations Number of ways of selecting r items out of n items Repetitions are not allowed Order is important Order is not important
- 22. How Do I Know Which One? Permutations Combinations Number of ways of selecting r items out of n items Repetitions are not allowed Order is important Order is not important Arrangements of r items from a set of n items Subsets of r items from a set of n items
- 23. How Do I Know Which One? Permutations Combinations Number of ways of selecting r items out of n items Repetitions are not allowed Order is important Order is not important Arrangements of r items from a set of n items Subsets of r items from a set of n items ( ) ( ) = − ! , ! n P n r n r ( ) ( ) = = − ! , ! ! n n C n r n r r r
- 24. How Do I Know Which One? Permutations Combinations Number of ways of selecting r items out of n items Repetitions are not allowed Order is important Order is not important Arrangements of r items from a set of n items Subsets of r items from a set of n items Clue words: arrangement, schedule, order Clue words: group, committee, sample, selection ( ) ( ) = − ! , ! n P n r n r ( ) ( ) = = − ! , ! ! n n C n r n r r r
- 25. How Do I Know Which One? ⚫ Caution: Not all counting problems lend themselves to either permutations or combinations. Whenever a tree diagram, addition principle, or multiplication principle can be used directly, use it.
- 26. Classwork ⚫ College Algebra 2e ⚫ 9.5: 8-32 (×4); 9.4: 30-44 (even); 9.3: 32-42 (even) ⚫ 9.5 Classwork Check ⚫ Quiz 9.4