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Counting Technique, Permutation, Combination
This document discusses counting techniques including permutations and combinations. It provides examples of using the fundamental counting principle to calculate the number of possible outcomes in situations like selecting shirts from various sizes and colors or generating license plates from letters and numbers. The key concepts of permutations, which involve ordered arrangements, and combinations, which involve unordered selections, are explained along with related formulas. Practice problems apply these techniques to scenarios such as seating arrangements, word rearrangements, and group selections.
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This section introduces counting techniques including the Fundamental Counting Principle, examples, and practice exercises.
This segment covers permutations including their definitions, factorial notation, and various examples explored.
Explains circular permutations, emphasizing arrangements of people and objects in circular formations.
Includes practice exercises and quizzes on permutations, arrangements, and combinations to assess understanding.
Discusses combinations with examples on selecting items from a set, emphasizing different selection scenarios.
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Counting Technique, Permutation, Combination
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- 4. Example 1 The ShirtMart sells shirts in sizes S, M, L, and XL. Each size comes in five colors: red, yellow, white, orange, and blue. The shirts come in short sleeve and long sleeve. How many kinds of shirts are there?
- 5. Example 2 A platenumber is made up of two consonants followed by three nonzero digits followed by a vowel. How many plate numbers are possible if a. The letters and digits cannot be repeated in the same plate number? b. The letters and digits can be repeated in the same plate number?
- 6. Example 3 Carla istaking a matching test in which he is supposed to match four answers with four questions. In how many different ways can he answer the four questions?
- 7. Practice Exercises 1. Howmany ordered pairs of letters are there that use only the letters A, B, C, D, and E? 2. How many different sequences of heads or tails are possible if a coin is flipped 8 times? 3. A model is selecting her outfits purchased 5 blouses, 4 skirts and 3 blazers. How many different new outfits consisting of a blouse, a skirt and a blazer can she create from her new collection? 4. Five different mathematics books and six different grammar books are to be arranged on a shelf. How many possible arrangements can be made if a. the books on the same subjects are to be arranged together? b. the books are to be arranged alternately? 5. From the word “ALERT,” determine how many letter arrangements are possible given the following conditions: a. All 5 letters are used without restrictions b. all vowels and consonants are together c. Only three letters are used without repeating any letter. d. Only four letters are used without restrictions.
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- 9. Factorial Notation In general,if n is a positive integer, then n factorial denoted by n! is the product of all integers less than or equal to n. n! = n. (n-1).(n-2). … . 2.1 As a special case, we define 0! = 1
- 10. Definition A permutation isthe ordered arrangement of distinguishable objects without allowing repetitions among the objects.
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- 12. Example 1 In howmany ways can a president and vice-president be chosen from a club with 12 members?
- 13. If there are50 floats in Penagbenga Festival, how many ways can a first-place, a second-place, and a third-place trophy be awarded?
- 14. Example 2 Find thenumber of different arrangments of the set of six letters HONEST a. Taken two at a time b. Taken three at a time c. Taken six at a time.
- 15. Example 3 Find thenumber of permutations in each situation. a. A softball coach chooses the first, second, and third batters for a team of 10 players. b. Three-digit numbers are formed from the digits 2, 3, 4, and 5, with no digits repeated.
- 16. Example 4 Five golferson a team are playing in a tournament. How many different line-up can the coach make?
- 17. Example 5 Find thenumber of ways a president, a vice-president, a secretary, and a treasurer can be chosen from among Alvin, Aris, Richard, Ricky, Alma, and Alice.
- 18. Permutations of Identicalobjects
- 19. Practice Exercises In howmany different ways can four people be seated in a row?
- 20. Example 1 In howmany ways may the letters of the word “STATISTICS” be arranged?
- 21. Example 2 In howmany ways may the letters of the word “ASSESSMENT” be arranged?
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- 23. Example In how manyways may seven persons be seated around a circular table?
- 24. Circular Permutations withno definite top or bottom
- 25. Example In how manyways can 3 keys be arranged in a key ring?
- 26. More exercises? Visit www.khanacademy.org, join the stat class 2013 and add me as coach (code: Q87N9A) for me to track your progress and give additional points.
- 27. Quiz 1. In howmany ways may the letters of the word ASSESSMENT be arranged? 2. How many license plates can be manufactured with three letters followed by three digits? No repetition 3. How many distinguishable ways can 4 beads be arranged in a circular bracelet? 4. In how many ways can 9 people be seated in a round table? 5. How many permutations of letters in the word GOOGOLPLEX?
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- 30. Example 1 Ellen receivedan offer to join a CD club. If she agrees to be a member, she can select 5 CDs from a list of 40 CDs. In how many ways can Ellen select the 5 CDs?
- 31. Example 2 Mr. Eltonhas to choose three of the six officers of the Math Club to go to a regional meeting. How many possible choices does he have?
- 32. Example 3 How manysubcommittees of 5 people can be formed from a committee consisting of 8 people?
- 33. Example 4 A classconsists of 12 boys and 15 girls. How many different committees of four can be selected from the class if each committee is to consist of two boys and two girls?
- 34. Practice Exercise 1. Howmany combinations of 5 records can be chosen from 12 records offered by a record club? 2. How many choices of 5 pocketbooks to read can be made from a set of nine pocketbooks? 3. A math professor gave his class a problem set consisting of 10 problems and required each student to answer any 7 problems. In how many ways can a student choose 7 problems from the problem set?