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0.5 – 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

0.5 Permutations & Combinations

0.5 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.Permutation Formula The number of permutations of n objects taken r at a time is the quotient of n! and (n r)!Permutation all possible arrangements of objects in which the order of the objects is taken in to consideration.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 all possible arrangements of objects in which the order of the objects is taken in to consideration.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)!Combinations a selection of objects in which order is not considered.

Permutation all possible arrangements of objects in which the order of the objects is taken in to consideration.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)!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!

Permutation all possible arrangements of objects in which the order of the objects is taken in to consideration.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)!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. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded? Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! P(10,3) = 10! (10 3)! Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! P(10,3) = 10! (10 3)! P(10,3) = 10! 7! Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! P(10,3) = 10! (10 3)! P(10,3) = 10! 7! P(10,3) = 10 9 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! P(10,3) = 10! (10 3)! P(10,3) = 10! 7! P(10,3) = 10 9 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! P(10,3) = 10! (10 3)! P(10,3) = 10! 7! P(10,3) = 10 9 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 P(10,3) = 10 9 8 Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?P(n,r) = n! (n r)! P(10,3) = 10! (10 3)! P(10,3) = 10! 7! P(10,3) = 10 9 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 P(10,3) = 10 9 8 = 720Ex. 2 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. 2 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. 2 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. 2 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. 2 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. 2 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

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!Ex. 3 How many different ways can the letters in the word MISSISSIPPI be arranged?

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!Ex. 3 How many different ways can the letters in the word MISSISSIPPI be arranged?11 total letters, 4 Is, 4 Ss, and 2 Ps.

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!Ex. 3 How many different ways can the letters in the word MISSISSIPPI be arranged?11 total letters, 4 Is, 4 Ss, and 2 Ps. n!_ p!q!

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!Ex. 3 How many different ways can the letters in the word MISSISSIPPI be arranged?11 total letters, 4 Is, 4 Ss, and 2 Ps. n!_ p!q! 11! _ 4!4!2!

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!Ex. 3 How many different ways can the letters in the word MISSISSIPPI be arranged?11 total letters, 4 Is, 4 Ss, and 2 Ps. n!_ p!q! 11! _ 4!4!2!11 10 9 8 7 6 5 4 3 2 1 4 3 2 1 4 3 2 1 3 2 1

Permutations with RepetitionThe number of permutations of n objects of which p are alike and q are alike is n!_ p!q!Ex. 3 How many different ways can the letters in the word MISSISSIPPI be arranged?11 total letters, 4 Is, 4 Ss, and 2 Ps. n!_ p!q! 11! _ 4!4!2!11 10 9 8 7 6 5 4 3 2 1 4 3 2 1 4 3 2 1 2 1

32511 5 3 7 5 4 3 2

11 5 3 7 5 4 3 = 34,650 2