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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. - PowerPoint PPT Presentation

12.4 Permutations & Combinations

12.4 Permutations & CombinationsPermutation 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)!

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!

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

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

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