Warm Up Melanie is choosing an outfit for a job interview. She has four dresses, three shirts, five pairs of pants and three pairs of shoes to choose.

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Permutations and Combinations

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Warm Up Melanie is choosing an outfit for a job interview. She has four dresses, three shirts, five pairs of pants and three pairs of shoes to choose from. How many possible outfits can Melanie wear?

Answer Melanie is choosing an outfit for a job interview. She has four dresses, three shirts, five pairs of pants and three pairs of shoes to choose from. How many possible outfits can Melanie wear? 57 outfits

Example 1 In how many different orders can 3 dogs line up to be groomed? In how many different orders can 10 dogs line up to be groomed?

Permutations

r selected items from a set of n items Permutations ORDER MATTERS! Two ways to solve: Fundamental Counting Principle This formula: r selected items from a set of n items This is “n” objects being taken “r” at a time where the ordering of “r” matters. P =

Example 2 – use Permutations: 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?

Sometimes, there isn’t a spot for every item Sometimes, there isn’t a spot for every item! For example, not every Olympian can get a medal. That’s okay. We handle it the same way.

Example 3 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?

Example 4: 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?

Permutations with Repetition Sometimes there are duplicate items in the set we are choosing from. Ex. In the word LOLLIPOP, there are three “L,” two “O,” and two “P” items in the set. The number of permutations of n items of which p are alike and q are alike is: Of note – if there are more than two alike items, you add more and more variables to the denominator – p!q!r!s! and so on.

Example 5 How many different ways can the letters of GEOMETRY be arranged? Since there are two E’s, repetition is a factor.

Example 6 There are 20 door prize winners at the banquet. One person gets a $100 gift card, 4 people get $25 gift cards, and 15 people get $5 gift cards. There are 100 people at the banquet who received tickets. In how many ways can the prizes be awarded?

Example 7 How many different ways can the letters of the word MATHEMATICS be arranged? There are a total of 11 letters