10.2 permutations

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10.2 Permutations 10.2 Permutations Objectives: Objectives: Solve problems involving Solve problems involving linear permutations of distinct or linear permutations of distinct or indistinguishable objects. Solve indistinguishable objects. Solve problems involving circular problems involving circular permutations. permutations. Standards: Standards: 2.7.8A Determine the 2.7.8A Determine the number of permutations for an event. number of permutations for an event.

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10.2 Permutations. Objectives: Solve problems involving linear permutations of distinct or indistinguishable objects. Solve problems involving circular permutations. Standards: 2.7.8A Determine the number of permutations for an event. A permutation is an arrangement of objects - PowerPoint PPT Presentation

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Page 1: 10.2 Permutations

10.2 Permutations10.2 PermutationsObjectives: Objectives: Solve problems involving linear Solve problems involving linear permutations of distinct or indistinguishable permutations of distinct or indistinguishable objects. Solve problems involving circular objects. Solve problems involving circular

permutations.permutations.

Standards: Standards: 2.7.8A Determine the number of 2.7.8A Determine the number of permutations for an event.permutations for an event.

Page 2: 10.2 Permutations

A permutation is an arrangement of objects in a specific order.

When objects are arranged in row, the permutation is called a linear permutation.

You can use factorial notation to abbreviate this product:

4! = 4 x 3 x 2 x 1 = 24. If n is a positive integer, then n factorial,

written n!, is defined as follows:n! = n x (n-1) x (n-2) x . . . x 2 x 1.

Note that the value of 0! = 1.

Page 3: 10.2 Permutations

I. Permutations of I. Permutations of n n Objects - the number ofObjects - the number ofpermutations of permutations of n n objects is given by objects is given by n!n!

{factorial button – go to Math to PRB to # 4}{factorial button – go to Math to PRB to # 4}

Ex 1. In 12-tone music, each of the 12 notes in an Ex 1. In 12-tone music, each of the 12 notes in an octave must be used exactly once before any are octave must be used exactly once before any are repeated. A set of 12 tones is called a tone row. repeated. A set of 12 tones is called a tone row. How many different tone rows are possible?How many different tone rows are possible?

Ex 2. How many different ways can the letters in the Ex 2. How many different ways can the letters in the word word objectsobjects be arranged? be arranged?

12! = 479,001, 600

7! = 5040

Page 4: 10.2 Permutations

II.II. Permutations of Permutations of n n Objects Taken Objects Taken r r at a Time – at a Time – the number of permutations of the number of permutations of n n

objects taken objects taken r r at a time, denoted by at a time, denoted by P(n, r)P(n, r), is , is given by given by P(n, r) = nPr P(n, r) = nPr ==____n!_n!_, where, where r r << n n. .

(n–r)!(n–r)!

Ex 1. Find the number of ways to listen to 5 different CDs from Ex 1. Find the number of ways to listen to 5 different CDs from a selection of 15 CDs. a selection of 15 CDs.

Ex 2. Find the number of ways to listen to 4 CDs from a Ex 2. Find the number of ways to listen to 4 CDs from a selection of 8 CDs.selection of 8 CDs.

Ex 3. Find the number of ways to listen to 3 different CDs from Ex 3. Find the number of ways to listen to 3 different CDs from a selection of 5 CDs.a selection of 5 CDs.

15 P 5 = 360, 360

8 P 4 = 1680

5 P 3 = 60

Page 5: 10.2 Permutations

III.III. Permutations with Identical Objects – Permutations with Identical Objects – the number of distinct permutations of the number of distinct permutations of

n n objects with objects with rr identical objects is identical objects is given by given by n!/r! n!/r! where where 1 1 << r r << n n. The . The

number of distinct permutations of number of distinct permutations of nn objectsobjects with with r1 r1 identical objects, identical objects, r2 r2 identical objects of another kind, identical objects of another kind, r3 r3

identical objects of another kind, . . . , identical objects of another kind, . . . , and and rk rk identical objects of another kind identical objects of another kind

is given byis given by______________nn! _ ! _ ..

r1 r1 ! * ! * r2 r2 ! * ! * r3 r3 ! . . . ! . . . rk rk !!

Page 6: 10.2 Permutations

Ex 1. Anna is planting 11 colored flowers in a line. In how many ways can she plant 4 red flowers,

5 yellow flowers, and 2 purple flowers?

Ex 2. In how many ways can Anna plant 11 colored flowers if 5 are white and the remaining ones are red?

11!__ (5! * 6!)

= 462

Page 7: 10.2 Permutations

Ex 3. Frank is organizing sports equipment for the physical education room. He has 15 balls that he must place in a line.

In how many ways can he line up 6 footballs, 2 soccer balls, 4 kickballs, and 3 basketballs?

Ex. 4 BETWEEN

____15!______(6! * 2! * 4! * 3!)

= 6,306,300

7!3! = 840

Page 8: 10.2 Permutations

III. Circular Permutations - If III. Circular Permutations - If nn distinct distinct objects are arranged around a circle, then objects are arranged around a circle, then

there are there are (n – 1)! (n – 1)! Circular permutations of Circular permutations of the the nn objects. objects.

Page 9: 10.2 Permutations

Ex 2. In how many ways can seats be chosen for 12 couples on a Ferris wheel that has 12 double seats?

Ex 3. In how many different ways can 17 students attending a seminar be arranged in a circular seating pattern?

(12 – 1)! = 11! = 39, 916, 800

(17 – 1)! = 16! = 2.09 X 1013

Page 10: 10.2 Permutations
Page 11: 10.2 Permutations

Writing ActivitiesWriting Activities

Page 12: 10.2 Permutations

REVIEW OF PERMUTATIONSREVIEW OF PERMUTATIONS