pencil, highlighter, GP notebook, textbook, calculator U11D3 Have out: Bellwork: Start IC – 34 on the worksheet. IC – 34 To choose a PIN (__________ _____________ ________) at a certain bank, you select 4 numbers from the digits 0 – 9, with no repeats. How many are possible? Make a decision chart to help you determine the answer. personal identification number total:

Factorials to Permutations IC – 34 Factorials to Permutations To choose a PIN (__________ _____________ ________) at a certain bank, you select 4 numbers from the digits 0 – 9, with no repeats. How many are possible? Make a decision chart to help you determine the answer. personal identification number +2 +2 10 9 8 7 _____ _____ _____ _____ = 5040 7 This looks like the beginning of 10!, but only down to _____, so the _____ is divided out. _____ _____ _____ _____ = 6! 10 9 8 7 So there are 4 spots to fill and 10 options to start with. How can I get a 6 out of this? 10 – 4 = 6 total:

You’ve won 14 trophies since grade school. Your mom says that IC – 35 You’ve won 14 trophies since grade school. Your mom says that only 3 trophies can go on the mantle. In how many ways can you choose and arrange them? 14 13 12 2184 _____ _____ _____ = 12 This looks like the beginning of 14!, but only down to _____ with _____ divided out. _____ _____ _____ = 11! 14 13 12 So there are 3 spots to fill and 14 options to start with. How can I get an 11 out of this? 14 – 3 = 11

Fifty–two women vie for the “Miss USA” crown. In how many IC – 36 Solve the following problems. Use factorials to express each of the following solutions. Fifty–two women vie for the “Miss USA” crown. In how many ways can a winner and her three runner–ups be chosen? 52 51 50 49 _____ _____ _____ _____ = 6,497,400 How can I write this as factorials? 52 – 4 = 48

Try b and c in your groups! b) The ECHS volleyball team is sponsoring a mixed–doubles sand court volleyball tournament and sixteen pairs have signed up for the chance to win one of the seven trophies and cash prizes. In how many ways can the teams be chosen and arranged for the top seven slots? Carmen is getting a new locker at school and the first thing she must do is decide on a new combination. The three number combination can be picked from the numbers 0 – 35. How many different locker combinations could she make up if none of the numbers can be repeated?

b) The ECHS volleyball team is sponsoring a mixed–doubles IC – 36 b) The ECHS volleyball team is sponsoring a mixed–doubles sand court volleyball tournament and sixteen pairs have signed up for the chance to win one of the seven trophies and cash prizes. In how many ways can the teams be chosen and arranged for the top seven slots? Remember, each pair is one entity. 16 15 14 13 12 11 10 _____ _____ _____ _____ _____ _____ _____ = 57,657,600 How can I write this as factorials? 16 – 7 = 9

Carmen is getting a new locker at school and the first thing she IC – 36 Carmen is getting a new locker at school and the first thing she must do is decide on a new combination. The three number combination can be picked from the numbers 0 – 35. How many different locker combinations could she make up if none of the numbers can be repeated? 36 35 34 42,840 _____ _____ _____= 36?! But it only went to 35! Oh, 0 – 35 is 36 different numbers. How can I write this as factorials? 36 – 3 = 33

PERMUTATIONS Per around perimeter periscope “_______” means ________, as in __________ or _________. Mutation change “__________” means __________. Problems where we are concerned about order and arrangements are called ________________. permutations For example, the 6 arrangements of the letters ABC are permutations. Purrrrrr! ABC CBA ACB CAB BCA BAC The two main characteristics of a permutation are _____ ____________ and __________ matters. NO REPETITION ORDER

There are two important questions we must ask from here on out. 1) Does order matter? 2) Are repeats allowed? Here is a flowchart to help you out. We will fill in the rest when we talk about combinations. Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES __________ (Leave blank) Combination Permutation Decision Chart

Practice: Determine whether or not the given situations are permutations or not. a) All the arrangements of the letters ABC. All the possible license plate letter triples that could be made using A, B, C. c) The possible 4–digit numbers you could make if you had square tiles, one with each number 2, 3, 4, 5, 6, 7, 8 The possible 4–digit numbers you could make if you could choose any digit from 2, 3, 4, 5, 6, 7, 8 e) From a group of 8 candidates, one will become president, one vice president, and one secretary of the school senate. f) From a group of 8 candidates, three will be selected to be on the executive committee.

Practice: Determine whether or not the given situations are permutations or not. a) All the arrangements of the letters ABC. All the possible license plate letter triples that could be made using A, B, C. Permutation Not a permutation Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart

Try c, d, e, and f in your groups! Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart

Practice: Determine whether or not the given situations are permutations or not. c) The possible 4–digit numbers you could make if you had square tiles, one with each number 2, 3, 4, 5, 6, 7, 8 The possible 4–digit numbers you could make if you could choose any digit from 2, 3, 4, 5, 6, 7, 8 Permutation Not a permutation Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart

Practice: Determine whether or not the given situations are permutations or not. e) From a group of 8 candidates, one will become president, one vice president, and one secretary of the school senate. f) From a group of 8 candidates, three will be selected to be on the executive committee. Not a permutation Permutation Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart

nPr PERMUTATION NOTATION selecting arranging nPr is the notation for _________ and _________ r things from n things. Add nPr to the flowchart. Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES ___________ (Leave blank) Combination Permutation Decision Chart nPr

things you are choosing from n = # of ______________________________ IC – 38 Eight people are running a race. In how many different ways can they come in first, second, and third? things you are choosing from n = # of ______________________________ r = # of ______________________________ things you are choosing 8 For this problem, n = ____ and r = ____ 3 _____ =

Try parts (b) and (c) on your own. Flip your worksheet over. Let’s go back and solve IC 34 – 36 using permutations this time. IC – 34 To choose a PIN at a certain bank, you select 4 numbers from the digits 0 – 9, with no repeats. How many are possible? We going to select and arrange 4 numbers from a group of 10 possible numbers. 10P4 = = 5040 IC – 36 Fifty–two women vie for the “Miss USA” crown. In how many ways can a winner and her three runner–ups be chosen? We going to select and arrange 4 women from a group of 52 women. 52P4 = = 6,497,400 Try parts (b) and (c) on your own.

b) The ECHS volleyball team is sponsoring a mixed–doubles IC – 36 b) The ECHS volleyball team is sponsoring a mixed–doubles sand court volleyball tournament and sixteen pairs have signed up for the chance to win one of the seven trophies and cash prizes. In how many ways can the teams be chosen and arranged for the top seven slots? We going to select and arrange 7 teams from a group of 16 possible teams. 16P7 = = 57,657,600 The three number combination can be picked from the numbers 0 – 35. How many different locker combinations could be made up if none of the numbers can be repeated? We going to select and arrange 3 numbers from a group of 36 possible numbers. 36P3 = = 42,840

Ice Cream Shop Try the Ice Cream problem in your groups!

You go to 31 flavors Ice Cream. Determine the number of ways you can order the following on an ice cream cone. a single scoop a double scoop (with repeats allowed) 31 31 31 = 961 Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart nPr

You go to 31 flavors Ice Cream. Determine the number of ways you can order the following on an ice cream cone. c) a double scoop (with no repeats) a triple scoop (with repeats allowed) 31 31 31 = 29,791 Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart nPr

You go to 31 flavors Ice Cream. Determine the number of ways you can order the following on an ice cream cone.. a triple scoop (with no repeats) Does order matter? NO YES Are repeats allowed? Are repeats allowed? NO YES NO YES Combination Permutation Decision Chart nPr

Finish the assignment: Log Practice and IC 40, 41, 43 – 48