12.4 – Permutations & Combinations
12.4 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.
Dec 24, 2015
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
12.4 – Permutations & Combinations
• Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.
• 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?
• 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) ∙ ∙ ∙ ∙
• 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 ∙ ∙ ∙ ∙
• 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∙ ∙ ∙ ∙
• 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 “!”.
• 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∙ ∙ ∙ ∙
Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)!
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)!
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?
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)!
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)!
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!
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 ∙ ∙ ∙
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 ∙ ∙ ∙
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∙ ∙ ∙ ∙ ∙
• Combinations – a selection of objects in which order is not considered.
• 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!
• 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!
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?
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!
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!
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 ∙ ∙ ∙ ∙ ∙ ∙ ∙
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 ∙ ∙ ∙ ∙ ∙ ∙ ∙
Related Documents
