Permutations, Combinations & Counting Principle

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Presentation transcript:

Permutations, Combinations & Counting Principle Essential Questions: What’s the difference between a permutation & combination? When do we use the counting principle? Can you find the total number of outcomes?

What is factorial and notation Factorial of a number means the product of the numbers starting with the number and going down to 1. Its only for positive integers. Its written Factorial of n is written as n! Examples: 5! = 5x4x3x2x1, 8!= 8x7x6…1

Permutation or Combination You Decide! My fruit salad contains grapes, pineapples, strawberries, and blueberries. Does it matter what order I place the ingredients in the bowl? Which one is it?

It is a Combination The order that you put the fruit in the fruit salad does not matter so you do not count the repeated events. So grapes, pineapple, strawberries, blueberries is the same as blueberries, strawberries, pineapple, grapes.

Permutation or Combination You Decide! My locker combination is 23-5-17. Does it matter what order I turn the numbers? Which one is it?

It is a permutation. Another set of numbers in a different order will not open your locker so the order DOES matter. So 23- 5 - 17 is very different that 23 - 17 - 5 or any other set of these 3 numbers.

Counting Principle – involves outcomes with different categories similar to a tree diagram. Example: You are making a password for your computer. You will use 2 letters and 1 number, repeats are allowed. You have different categories: Letter • Letter • Number 26 • 26 • 10 = 6760 outcomes possible

YOU TRY! How many ways can 7 students finish a race in 1st , 2nd, and 3rd place? First decide if the order matters or not. Then calculate.

2) How many ways can you order a white, chocolate, or yellow cake, with chocolate or vanilla icing, and 20 possible designs on top? First think about what kind of problem this is. Are there different categories or not? Does the order matter?

2) This uses the counting principle because there are 3 different categories involved: the type of cake, type of icing, and the type of design. 3 • 2 • 20 = 120 possible outcomes

How many ways can you arrange 3 sweaters in a display window from 8 sweaters? First decide whether the order matters. Then calculate.

Permutation (counting principle) example without replacement. You are creating a password with 4 digits. You cannot repeat a number, how many possible arrangements are there? There are 10 digits, so start with that and each other digit will go down by one. So 10 • 9 • 8 • 7 = 5040 possible outcomes