12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which...
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Transcript of 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which...
![Page 1: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/1.jpg)
12.4 – Permutations & Combinations
![Page 2: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/2.jpg)
• Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.
![Page 3: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/3.jpg)
• 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?
![Page 4: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/4.jpg)
• 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) ∙ ∙ ∙ ∙
![Page 5: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/5.jpg)
• 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 ∙ ∙ ∙ ∙
![Page 6: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/6.jpg)
• 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∙ ∙ ∙ ∙
![Page 7: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/7.jpg)
• 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 “!”.
![Page 8: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/8.jpg)
• 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∙ ∙ ∙ ∙
![Page 9: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/9.jpg)
Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)!
![Page 10: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/10.jpg)
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)!
![Page 11: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/11.jpg)
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?
![Page 12: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/12.jpg)
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)!
![Page 13: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/13.jpg)
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)!
![Page 14: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/14.jpg)
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!
![Page 15: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/15.jpg)
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 ∙ ∙ ∙
![Page 16: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/16.jpg)
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 ∙ ∙ ∙
![Page 17: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/17.jpg)
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∙ ∙ ∙ ∙ ∙
![Page 18: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/18.jpg)
• Combinations – a selection of objects in which order is not considered.
![Page 19: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/19.jpg)
• 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!
![Page 20: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/20.jpg)
• 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!
![Page 21: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/21.jpg)
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?
![Page 22: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/22.jpg)
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!
![Page 23: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/23.jpg)
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!
![Page 24: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/24.jpg)
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 ∙ ∙ ∙ ∙ ∙ ∙ ∙
![Page 25: 12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649dbb5503460f94aac8a6/html5/thumbnails/25.jpg)
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 ∙ ∙ ∙ ∙ ∙ ∙ ∙