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Integrated Math 2 Section 4-6
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Integrated Math 2 Section 4-6

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Permutations of a Set

Permutations of a Set

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  • Transcript

    • 1. Section 4-6 Permutations of a Set
    • 2. Essential Question How do you find the number of permutations of a set? Where you’ll see this: Cooking, travel, music, sports, games
    • 3. Vocabulary 1. Permutation: 2. Factorial: 3. n Pr :
    • 4. Vocabulary 1. Permutation: An arrangement of items in a particular order 2. Factorial: 3. n Pr :
    • 5. Vocabulary 1. Permutation: An arrangement of items in a particular order ORDER IS IMPORTANT!!! 2. Factorial: 3. n Pr :
    • 6. Vocabulary 1. Permutation: An arrangement of items in a particular order ORDER IS IMPORTANT!!! 2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)... (3)(2)(1) 3. n Pr :
    • 7. Vocabulary 1. Permutation: An arrangement of items in a particular order ORDER IS IMPORTANT!!! 2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)... (3)(2)(1) 5! = (5)(4)(3)(2)(1) 3. n Pr :
    • 8. Vocabulary 1. Permutation: An arrangement of items in a particular order ORDER IS IMPORTANT!!! 2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)... (3)(2)(1) 5! = (5)(4)(3)(2)(1) = 120 3. n Pr :
    • 9. Vocabulary 1. Permutation: An arrangement of items in a particular order ORDER IS IMPORTANT!!! 2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)... (3)(2)(1) 5! = (5)(4)(3)(2)(1) = 120 3. n Pr : n permutations taken r at a time; n is the number of different items and r is the number of items taken at a time
    • 10. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
    • 11. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
    • 12. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed? 3
    • 13. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed? 3 2
    • 14. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed? 3 2 1
    • 15. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed? 3 2 1 (3)(2)(1)
    • 16. Example 1 How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed? 3 2 1 (3)(2)(1) = 6 numbers
    • 17. Permutation Formula
    • 18. Permutation Formula n! n Pr = (n − r)!
    • 19. Permutation Formula n! n Pr = (n − r)! n is the number of different items and r is the number of items taken at a time
    • 20. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
    • 21. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)!
    • 22. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4 P2
    • 23. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! P = 4 2 (4 − 2)!
    • 24. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! 4! P = 4 2 = (4 − 2)! 2!
    • 25. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! 4! (4)(3)(2)(1) P = 4 2 = = (4 − 2)! 2! (2)(1)
    • 26. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! 4! (4)(3)(2)(1) P = 4 2 = = (4 − 2)! 2! (2)(1)
    • 27. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! 4! (4)(3)(2)(1) P = 4 2 = = (4 − 2)! 2! (2)(1)
    • 28. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! 4! (4)(3)(2)(1) P = 4 2 = = =12 (4 − 2)! 2! (2)(1)
    • 29. Example 2 Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters. n! P= n r (n − r)! 4! 4! (4)(3)(2)(1) P = 4 2 = = =12 (4 − 2)! 2! (2)(1) 12 two-letter groupings can be made
    • 30. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
    • 31. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)!
    • 32. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)! 6 P3
    • 33. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)! 6! P = 6 3 (6−3)!
    • 34. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)! 6! 6! P = 6 3 = (6−3)! 3!
    • 35. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)! 6! 6! P = 6 3 = = (6)(5)(4) (6−3)! 3!
    • 36. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)! 6! 6! P = 6 3 = = (6)(5)(4) =120 (6−3)! 3!
    • 37. Example 3 How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed? n! P= n r (n − r)! 6! 6! P = 6 3 = = (6)(5)(4) =120 (6−3)! 3! 120 3-digit number can be formed
    • 38. Understanding Break 1. What is the short way of writing (6)(5)(4)(3)(2)(1)? 2. Is (2!)(3!) the same as 6! ? 3. Which of the following is the value of 0! ? a. 0 b. 1 c. -1 d. ±1 4. Which of the following are examples of permutations? a. Arranging 5 people on a bench that seats 5 b. Choosing 4 letters from 6 and writing all possible arrangements of the choices c. Selecting a committee of 3 from 10 women
    • 39. Understanding Break 1. What is the short way of writing (6)(5)(4)(3)(2)(1)? 6! 2. Is (2!)(3!) the same as 6! ? 3. Which of the following is the value of 0! ? a. 0 b. 1 c. -1 d. ±1 4. Which of the following are examples of permutations? a. Arranging 5 people on a bench that seats 5 b. Choosing 4 letters from 6 and writing all possible arrangements of the choices c. Selecting a committee of 3 from 10 women
    • 40. Understanding Break 1. What is the short way of writing (6)(5)(4)(3)(2)(1)? 6! 2. Is (2!)(3!) the same as 6! ? No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1) 3. Which of the following is the value of 0! ? a. 0 b. 1 c. -1 d. ±1 4. Which of the following are examples of permutations? a. Arranging 5 people on a bench that seats 5 b. Choosing 4 letters from 6 and writing all possible arrangements of the choices c. Selecting a committee of 3 from 10 women
    • 41. Understanding Break 1. What is the short way of writing (6)(5)(4)(3)(2)(1)? 6! 2. Is (2!)(3!) the same as 6! ? No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1) 3. Which of the following is the value of 0! ? a. 0 b. 1 c. -1 d. ±1 4. Which of the following are examples of permutations? a. Arranging 5 people on a bench that seats 5 b. Choosing 4 letters from 6 and writing all possible arrangements of the choices c. Selecting a committee of 3 from 10 women
    • 42. Understanding Break 1. What is the short way of writing (6)(5)(4)(3)(2)(1)? 6! 2. Is (2!)(3!) the same as 6! ? No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1) 3. Which of the following is the value of 0! ? a. 0 b. 1 c. -1 d. ±1 4. Which of the following are examples of permutations? a. Arranging 5 people on a bench that seats 5 b. Choosing 4 letters from 6 and writing all possible arrangements of the choices c. Selecting a committee of 3 from 10 women
    • 43. Understanding Break 1. What is the short way of writing (6)(5)(4)(3)(2)(1)? 6! 2. Is (2!)(3!) the same as 6! ? No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1) 3. Which of the following is the value of 0! ? a. 0 b. 1 c. -1 d. ±1 4. Which of the following are examples of permutations? a. Arranging 5 people on a bench that seats 5 b. Choosing 4 letters from 6 and writing all possible arrangements of the choices c. Selecting a committee of 3 from 10 women
    • 44. Homework
    • 45. Homework p. 174 #1-25 odd "The inner fire is the most important thing mankind possesses." - Edith Sodergran