Permutations and Combinations AII.12 2009

Bell Work Find the number of possible outcomes 1. bagels: plain, egg, wheat, onion meat: turkey, ham, roast beef, tuna 2. eggs: scrambled, over easy, hard boiled meat: sausage patty, sausage link, bacon, ham

Objectives: apply fundamental counting principle compute permutations compute combinations distinguish permutations vs combinations

Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more characteristics . Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m* n possible outcomes for the two events together.

Fundamental Counting Principle I own 8 shirts and 5 pairs of shorts. How many possible outfits can I choose from? 8*5 = 40 outfits

Fundamental Counting Principle For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? 4*3*2*5 = 120 outfits

To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there? Yogurt Parfait (choose 1 of each) Flavor Plain Vanilla Fruit Peaches Strawberries Bananas Raspberries Blueberries Nuts Almonds Peanuts Walnuts

number of flavors number of fruits number of nuts number of choices times times equals 2  5  3 = 30 There are 30 parfait choices.

How many different ways can I get to the 3rd Floor??

Permutations Notice, ORDER MATTERS! A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS! To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

Permutations The number of ways to arrange the letters ABC: ____ ____ ____ 3 ____ ____ Number of choices for first blank? 3 2 ___ Number of choices for second blank? Number of choices for third blank? 3 2 1 3*2*1 = 6 ABC ACB BAC BCA CAB CBA

Permutations From a club of 8 members, a President, Vice President, Secretary, Treasurer are to be elected. In how many ways can the offices be filled? P VP S T

Permutations The number of ways to arrange the letters LMNOP

Permutations In how many ways can you arrange 3 out of 5 books on a shelf?

Combinations ORDER DOES NOT MATTER! A Combination is an arrangement of items in which order does not matter. ORDER DOES NOT MATTER! Since the order does not matter in combinations, there are fewer combinations than permutations.  The combinations are a "subset" of the permutations.

Combinations In how many ways can you choose 2 different side dishes from a menu containing 5 items?

Combinations How many different groups of 3 ice cream toppings can you choose from 10 toppings?

Mini Quiz 1. A group of 8 swimmers are swimming in a race. Prizes are given for first, second, and third place. How many different outcomes can there be? 2. A lunch special includes one main item, one side, and one drink. 3. When ordering a pizza, you can choose 2 toppings from the following: mushrooms, olives, pepperoni, pineapple, and sausage. How many different types of pizza can you order?

4. There are 8 hot air balloons in a race 4. There are 8 hot air balloons in a race. In how many possible orders can all 8 hot air balloons finish the race? 5. A group of 12 people are forming a committee. How many different 4-person committees can be formed BONUS: How many different ways can you arrange a baseball batting order?