Permutations Combinations PERMUTATIONS Permutations Arrangement of items in
Permutations & Combinations PERMUTATIONS
Permutations �Arrangement of items in a particular order �Can be computed using three different methods ◦ Factorials ◦ Multiplication Counting Principle ◦ n. Pr function on the calculator �Method used depends upon situation involved
Factorials For any positive integer n, n! = n(n − 1) • . . . • 3 • 2 • 1 For n = 0, n! = 1 Example: 6! = 6 • 5 • 4 • 3 • 2 • 1
Example In how many different orders can ten dogs line up to be groomed? �Can use Multiplication Counting Principle or Factorial
Solution �Since all ten dogs are being groomed, use all items from the original set �Ten ways to select the first dog in line, nine ways to select the next dog, and so on 10 • 9 • . . . • 2 • 1 = 10! = 3, 628, 800 �The ten dogs can line up in 3, 628, 800 different orders
Your Turn… �In how many ways can you arrange six trophies on a shelf? �In how many ways can four tires be arranged on a car? �If the spare tire is included, how many ways can the tires be arranged on a car?
Solutions �In how many ways can you arrange six trophies on a shelf? ◦ 6! = 720 �In how many ways can four tires be arranged on a car? ◦ 4! = 24 �If the spare tire is included, how many ways can the tires be arranged on a car? ◦ 5! = 120
Permutations �Some permutations do not use all of the items available in a set �These permutations may allow or disallow repetition/replacement �The number or permutations of n items of a set arranged r items at a time is P=
Example �Seven yachts enter a race. �First, second and third places will be given to the three fastest yachts. �How many arrangements of first, second, and third places are possible with seven yachts?
Solution �Using Multiplication Counting Principle ◦ 7 • 6 • 5 = 210 �Using the Permutation Formula ◦ 7 yachts so n = 7 ◦ Arranged 3 at a time so r = 3 P = 7!/(7 -3)! P = 7!/4! P = 5040/24 P = 210
Your Turn… �Fifteen applicants want to interview with SAS. In how many ways can the 10 time slots be assigned? �How many different nine-player batting orders can be chosen from a baseball squad of 16? �There are 10 finalists in an archery competition. How many ways can the gold, silver & bronze medals be awarded?
Solutions �Fifteen applicants want to interview with SAS. In how many ways can the 10 time slots be assigned? 1. 0897 x 1010 �How many different nine-player batting orders can be chosen from a baseball squad of 16? 4, 151, 347, 200 �There are 10 finalists in an archery competition. How many ways can the gold, silver & bronze medals be awarded?
Permutations with repetition �Repetitions must be considered �The number of permutations of n items of which p are alike and q are alike is
Example �How many different ways can the letters of GEOMETRY be arranged? �Since there are two E’s, repetition is a factor…
Example �How many different ways can the letters of the word MISSISSIPPI be arranged? �There a total of 11 letters �The following letters are repeated ◦ S is repeated 4 times ◦ I is repeated 4 times ◦ P is repeated 2 times
Solution �
Example �How many different ways can the letters of the word MATHEMATICS be arranged? �There a total of 11 letters �The following letters are repeated ◦ M is repeated 2 times ◦ A is repeated 2 times ◦ T is repeated 2 times
Solution
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