12.4 – Permutations & Combinations

1. 12.4 – Permutations & Combinations

2. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.

3. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration. Ex. 1 A travel agency is planning a vacation package in which travelers will visit 5 cities around Europe. How many ways can the agency arrange the 5 cities along the tour?

4. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration. Ex. 1 A travel agency is planning a vacation package in which travelers will visit 5 cities around Europe. How many ways can the agency arrange the 5 cities along the tour? 5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4)

5. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration. Ex. 1 A travel agency is planning a vacation package in which travelers will visit 5 cities around Europe. How many ways can the agency arrange the 5 cities along the tour? 5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4) 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1

6. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration. Ex. 1 A travel agency is planning a vacation package in which travelers will visit 5 cities around Europe. How many ways can the agency arrange the 5 cities along the tour? 5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4) 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120

7. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration. Ex. 1 A travel agency is planning a vacation package in which travelers will visit 5 cities around Europe. How many ways can the agency arrange the 5 cities along the tour? 5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4) 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120 *This is called factorial, represented by “!”.

8. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration. Ex. 1 A travel agency is planning a vacation package in which travelers will visit 5 cities around Europe. How many ways can the agency arrange the 5 cities along the tour? 5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4) 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120 *This is called factorial, represented by “!”. 5! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120

9. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)!

10. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)!

11. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case?

12. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case? P(n,r) = n! (n – r)!

13. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case? P(n,r) = n! (n – r)! P(10,6) = 10! (10 – 6)!

14. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case? P(n,r) = n! (n – r)! P(10,6) = 10! (10 – 6)! P(10,6) = 10! 4!

15. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case? P(n,r) = n! (n – r)! P(10,6) = 10! (10 – 6)! P(10,6) = 10! 4! P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 4 ∙ 3 ∙ 2 ∙ 1

16. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case? P(n,r) = n! (n – r)! P(10,6) = 10! (10 – 6)! P(10,6) = 10! 4! P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 4 ∙ 3 ∙ 2 ∙ 1

17. Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)! P(n,r) = n! (n – r)! Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a showcase. How many ways can she arrange the magazines in the case? P(n,r) = n! (n – r)! P(10,6) = 10! (10 – 6)! P(10,6) = 10! 4! P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 4 ∙ 3 ∙ 2 ∙ 1 P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 = 151,200

18. Combinations – a selection of objects in which order is not considered.

19. Combinations – a selection of objects in which order is not considered. Combination Formula – The number of combinations of n objects taken r at a time is the quotient of n! and (n – r)!r!

20. Combinations – a selection of objects in which order is not considered. Combination Formula – The number of combinations of n objects taken r at a time is the quotient of n! and (n – r)!r! C(n,r) = n! (n – r)!r!

21. Ex. 3 Horatio works part-time at a local department store. His manager asked him to choose for display 5 different styles of shirts from the wall of the store that has 8 shirts on it to put in a display. How many ways can he choose the shirts?

22. Ex. 3 Horatio works part-time at a local department store. His manager asked him to choose for display 5 different styles of shirts from the wall of the store that has 8 shirts on it to put in a display. How many ways can he choose the shirts? C(n,r) = n! (n – r)!r!

23. Ex. 3 Horatio works part-time at a local department store. His manager asked him to choose for display 5 different styles of shirts from the wall of the store that has 8 shirts on it to put in a display. How many ways can he choose the shirts? C(n,r) = n! (n – r)!r! C(8,5) = 8! (8 – 5)!5!

24. Ex. 3 Horatio works part-time at a local department store. His manager asked him to choose for display 5 different styles of shirts from the wall of the store that has 8 shirts on it to put in a display. How many ways can he choose the shirts? C(n,r) = n! (n – r)!r! C(8,5) = 8! (8 – 5)!5! C(8,5) = 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 3 ∙ 2 ∙ 1 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1

25. Ex. 3 Horatio works part-time at a local department store. His manager asked him to choose for display 5 different styles of shirts from the wall of the store that has 8 shirts on it to put in a display. How many ways can he choose the shirts? C(n,r) = n! (n – r)!r! C(8,5) = 8! (8 – 5)!5! C(8,5) = 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 56 3 ∙ 2 ∙ 1 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1