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.

• 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 ∙ ∙ ∙ ∙ ∙ ∙ ∙