pencil, highlighter, GP notebook, calculator, red pen U11D4 Have out: Bellwork: You go to a store called “40 Flavors.” In how many ways can you order: a) a two scoop cone (with repeats) b) two scoop cone (without repeats) +2 ______ = 40 40 1600 +2 +1 +1 c) 3 scoop cone (with repeats) d) 3 scoop cone (without repeats) +2 _________ = 40 40 40 64,000 +2 +1 total: +1

Permutations and Combinations Recall: Permutations – __________ matters and no ____________. order repetition nPr is choosing and arranging ___ things from ___ things. r n Example #1: You go to a store called “27 Flavors.” In how many ways can you order: a) a two scoop cone (with repeats) b) two scoop cone (without repeats) ______ = 27 27 729

c) 3 scoop cone (with repeats) d) 3 scoop cone (without repeats) _________ = 27 27 27 19,683

e) A two scoop bowl (without repeats) Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Leave blank Permutation Decision Chart According to our flowchart, this is a combination. So how do we figure this one out?

Part e: There is no ________ in a bowl. This is called a _____________. In how many ways can you order a two scoop bowl (without repeats)? order combination This is NOT a permutation. If it was, how would you calculate the possibilities? But since order does NOT matter, we have repeats. This is how to get rid of the repeats: Will it always be 2! ? Where did the two come from? Let us look at an example with less possibilities.

Example #2: Amy, Betty, Carla, Danielle, Erin, and Francis all try out for 2 places on the cheer squad. Suppose there were two positions (cheer leader and co–leader) that the girls were trying out for. How many different ways can the positions be filled? This is a permutation! b) Suppose that the girls were going to be on a cheer committee. Write out all the possible ways to choose 2 different girls. (Be methodical!) This is a combination! AB BC CD DE EF What about BA, CA . . . ? AC BD CE DF These don’t matter since order doesn’t matter. AD BE CF AE BF 15 # of combinations: ______ AF

Since order does not matter, this is not a ____________ Since order does not matter, this is not a ____________. In part (b), we care about who is selected but we do not care about the order of the selection or any arrangement of the groups. Selections of committees, or of subsets of items from a larger set without regard to the order of the group selected, are called ______________. permutation combinations In example 2 part b, # of permutations # of combinations = _________________ = ____ = _____ = ____ # of arrangements

To win the California lottery, a person must choose the correct IC – 51 To win the California lottery, a person must choose the correct ____________ of numbers; that is, choosing the correct 6 of 51 numbers. (_________ does not matter.) combination order How many combinations of 6 numbers can be chosen from the list of 51? Total # of ways to arrange 6 #s out of 51 #s when order matters. # of combinations = _______ = _______________ = _________ How many ways are there to arrange 6 numbers? What is the probability of winning the lottery by choosing just 1 set of 6 numbers?

Complete IC – 52 in your groups. Five cards are drawn from a standard deck of 52 cards. Use permutation notation to calculate the number of choices for the 1st card, 2nd card, and so forth up to the fifth card. How many ways can you arrange the five cards selected? Write your answer both as a number and using a factorial. Since order generally does not matter when playing cards, we need to divide out the number of repetitions of the same set of five cards. Calculate the number of five card hands that can be selected from a deck of 52 cards. # of combinations =

Five cards are drawn from a standard deck of 52 cards. IC – 52 Five cards are drawn from a standard deck of 52 cards. Use permutation notation to calculate the number of choices for the 1st card, 2nd card, and so forth up to the fifth card. How many ways can you arrange the five cards selected? Write your answer both as a number and using a factorial. Since order generally does not matter when playing cards, we need to divide out the number of repetitions of the same set of five cards. Calculate the number of five card hands that can be selected from a deck of 52 cards. # of combinations =_______ = ____________ = ___________

Let’s summarize what we have learned thus far… IC – 53 Let’s summarize what we have learned thus far… arrange Factorials  These count the number of ways to ________ a group of objects in ____________. order choose arrange Permutations  These count ways to ________ and _________ a subgroup from a larger group. ______ matters! These are based on decision charts: how many ways to pick the first object, times how many ways to pick the second, etc. order choosing arranging nPr is __________ and ___________ r things from n things. n! (n – r)! Combinations  These count ways to just ________ a subgroup from a larger group. choose # of ways to choose and arrange # of combinations = _____________________________ # of ways to arrange

COMBINATIONS choosing nCr represents __________ r things from n things. This is when _______ does NOT matter. order Note: There are __ times as many permutations as combinations. r! Remember, you divide by r! because you are getting rid of the repeats from the permutation. They are now repeats because with combinations, order does NOT matter.

nCr nPr nr Add to the flow chart: Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart nCr nPr nr

Examples: a) Ten people are running for the prom committee. In how many different ways can a committee of 4 be chosen? = 210 b) You go to an ice cream store called “29 Flavors.” How many different ways can you order a 5 scoop bowl (without repeats)? = 118,755

Examples: c) Thirty employees are running for a union committee. In how many different ways can a committee of 20 be chosen? = 30,045,015 d) Cody has 15 hard boiled eggs. How many different ways can he select 6 eggs to decorate for a project? = 5005

Finish the assignment: IC 54 – 60, 63, 64, 66

old bellwork

pencil, highlighter, GP notebook, calculator, red pen 3/18/11 Have out: Bellwork: 1. How many different 4 letter passwords can you have if you can not have repeats and you can only use the letters g through p? total:

Bellwork: g, h, I, j, k, l, m, n, o, p There are 5040 passwords. How many different 4 letter passwords can you have if you can not have repeats and you can only use the letters g through p? Bellwork: g, h, I, j, k, l, m, n, o, p +2 +2 There are 5040 passwords. total: