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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.

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Presentation on theme: "0.5 – Permutations & Combinations. Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration."— Presentation transcript:

1 0.5 – Permutations & Combinations

2 Permutation – all possible arrangements of objects in which the order of the objects is taken in to consideration.

3 Permutation Formula – The number of permutations of n objects taken r at a time is the quotient of n! and (n – r)!

4 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)!

5 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.

6 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!

7 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!

8 Ex. 1 There are 10 finalist in an Olympic competition. How many different ways can gold, silver, & bronze medals be awarded?

9 P(n,r) = n! (n – r)!

10 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)!

11 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!

12 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) =

13 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) =

14 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) = P(10,3) =

15 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) = P(10,3) = = 720

16 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?

17 C(n,r) = n! (n – r)!r!

18 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!

19 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) =

20 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) =

21 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) = =

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

23 Permutations with Repetition The 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?

24 Permutations with Repetition The 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.

25 Permutations with Repetition The 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!

26 Permutations with Repetition The 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!

27 Permutations with Repetition The 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!

28 Permutations with Repetition The 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!

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30 = 34,650 2


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