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# Permutations

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### Permutations

1. 1. Quiz 6 You may answer each item as a product of factors (e.g. 4 · 3 · 2) 1 How many 4-digit whole numbers can be formed if there is no repetition of digits? 2 How many 4-digit odd whole numbers can be formed if repetition of digits is allowed? 3 How many whole numbers less than 10000 can be formed using odd digits? 4 How many 5-digit whole numbers can be formed if odd and even digits should alternate and there is no repetition of digits? 5 How many 2-digit whole numbers can be formed that are not perfect squares?Mathematics 4Permutations
2. 2. Quiz 7 You may answer each item as a product of factors (e.g. 4 · 3 · 2) 1 How many ways can you arrange the letters in the word CESIUM? 2 In a class of 15 students, how many ways can a president, vice president, and a secretary be chosen? 3 How many ways can 10 people be lined up if 4 of them always have to be together? 4 How many ways can 5 prom couples be lined up if each couple must be together? 5 How many ways can the same 5 prom couples be arranged in a line if each girl is between her date and another girl, or beside her date only?Mathematics 4Permutations
3. 3. Permutations Mathematics 4 February 22, 2012Mathematics 4Permutations
4. 4. Deﬁnition of permutations Permutation A permutation is an arrangement of all or part of a set of objectsMathematics 4Permutations
5. 5. Permutation formulas Linear Permutation The number of permutations of n distinct objects taken r at a time is n! n Pr = (1) (n − r)!Mathematics 4Permutations
6. 6. Example 1 How many three-letter ”words” can you form from the word CHEMISTRY?Mathematics 4Permutations
7. 7. Example 2 In how many ways can 8 people line up to get on a bus: a. if three speciﬁc persons insist on following each other?Mathematics 4Permutations
8. 8. Example 2 In how many ways can 8 people line up to get on a bus: a. if three speciﬁc persons insist on following each other? b. if two speciﬁc persons refuse to follow each other?Mathematics 4Permutations
9. 9. Example 3 How many ﬁve-letter ”words” can you form from the letters A, E, I, O, U, B, C, D, F, G, H, J, with no repetition of letters: a. if vowels and consonants must alternate?Mathematics 4Permutations
10. 10. Example 3 How many ﬁve-letter ”words” can you form from the letters A, E, I, O, U, B, C, D, F, G, H, J, with no repetition of letters: a. if vowels and consonants must alternate? b. if each word must start and end with a vowel?Mathematics 4Permutations
11. 11. Permutation formulas Distinguishable Permutations The number of permutations of n things of which n1 are of one kind, n2 of a second kind, ..., nk of a kth kind is n! (2) n1 !n2 !...nk !Mathematics 4Permutations
12. 12. Example 4 How many ways can you arrange the letters in the word MATHEMATICS?Mathematics 4Permutations
13. 13. Example 5 How many ways can 3 rose bushes, 4 santan bushes, and 2 gumamela bushes be arranged along a property line if one does not distinguish between bushes of the same kind?Mathematics 4Permutations
14. 14. Permutation formulas Circular Permutations The number of permutations of n distinct objects arranged in a circle is (n − 1)! (3)Mathematics 4Permutations
15. 15. Example 6 How many ways can 5 couples be seated around a table a. if there are no restrictions?Mathematics 4Permutations
16. 16. Example 6 How many ways can 5 couples be seated around a table a. if there are no restrictions? b. if each couple is to seat together?Mathematics 4Permutations
17. 17. Example 6 How many ways can 5 couples be seated around a table a. if there are no restrictions? b. if each couple is to seat together? c. if each couple is to seat across each other?Mathematics 4Permutations
18. 18. Example 6 How many ways can 5 couples be seated around a table a. if there are no restrictions? b. if each couple is to seat together? c. if each couple is to seat across each other? d. if men and women alternate?Mathematics 4Permutations
19. 19. Permutation formulas Ring Permutations The number of permutations of n distinct objects arranged in a circle with no distinction for clockwise or counterclockwise is (n − 1)! (4) 2Mathematics 4Permutations
20. 20. Example 7 How many ways can 7 keys be arranged on a key ring a. if there are no restrictions?Mathematics 4Permutations
21. 21. Example 7 How many ways can 7 keys be arranged on a key ring a. if there are no restrictions? b. if two keys must always be together?Mathematics 4Permutations
22. 22. Deﬁnition of combinations Combination A combination is selecting objects with no regard to orderMathematics 4Permutations
23. 23. Combination formula Combination The number of combinations of n distinct objects taken r at a time is n Pr n! n Cr = = (5) r! (n − r)!r!Mathematics 4Permutations
24. 24. Example 8 How many teams of 3 students from your class be selected for the team category of the Math intersection?Mathematics 4Permutations
25. 25. Example 9 How many subsets of a set of 10 elements have either 3 or 4 elements?Mathematics 4Permutations
26. 26. Example 10 How many diﬀerent committees of 4 to 6 persons be chosen from a society of 28 members?Mathematics 4Permutations
27. 27. Example 11 How many ways can a team of 3 boys and 4 girls be chosen from your class?Mathematics 4Permutations