Publish on 11th November 2019
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Probability Theory--Counting: Multiplication, Permutations, Combinations

104- Ch. 11 part 1; 260- Ch. 5 pt.2master_prob_pc.pptx

Permutation Examples

1. If there are 4 people in the math club (Anne,Bob,Cindy,Dave), and we wish to elect a president and vice-president, LIST all of the different ways that this is possible.2. From these 4 people (Anne,Bob,Cindy,Dave), we wish to elect a president, vice-president, and treasurer. LIST all of the different ways that this is possible.

Answers

1. If there are 4 people in the math club (Anne,Bob,Cindy,Dave), and we wish to elect a president and vice-president, LIST all of the different ways that this is possible.AB BA CA DAAC BC CB DBAD BD CD DC4*3=12 or 4P2 = 12

A B C ABCD ABDC B ACBD ACDD A BDAC BDCB A C BACD BCDC A BCAD BCDD A BDAC BDCC A B CABD CADB A CBAD CBDA B DABC DACD A B DABC DACB A DBAC DBCC A DCAB DCB4*3*2 = 24 outcomesOr 4P3 = 24

More counting examples:

1. At a restaurant, you have a choice of main dish (beef, chicken, fish), vegetable (broccoli, corn), potato (baked, fries), and dessert (chocolate, strawberry). LIST all possible choices.How many choices are there?__

2. And 3.

2. A car comes in 6 colors, with or without air conditioning, and with standard or automatic.How many different designs are there?3. Pants are available in 5 colors, 6 sizes, and 3 lengths. How many different types are there?

4. And 5. T/F quiz

4. A teacher wishes to make all possible different answer keys to a T/F quiz to cut down on cheating. How many possible different answer keys could there be if there are 4 questions?5. What if there were 10 T/F questions.

6.And 7. Multiplechoice test

Ateacher wishes to make all possible different answer keys to a multiple choice quiz. How many possible different answer keys could there be if there are 3 questions that each have 4 choices (A,B,C,D)?(You can usea tree diagram to list them.)What if there were 20 multiple choice questions with 5 choices each?

8. And 9. Batting orders

8. With 9 baseball players on a team, how many different batting orders exist?9. With 12 baseball players on a team, how many different batting orders exist?

Summary of Counting Rules

Fundamental Counting Principle -- Multiplication RuleIf you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices is M*N.Permutation of n things taken r at a timenPr= n!/(n-r)!In permutations, ORDER matters & REPETITION is NOT allowed?

For each, decide which type of problem:

1. If you have 4 flags, how many different ways could you raise all of them on a flagpole?2. If you have 7 flags, how many different ways could you raise 4?3. If license plates have 3 letters and then 4 numbers, how many different license plates exist?

…decidewhich type of problem:

4. With 9 possible speakers, how many ways could an ordered list of 6 be decided?5. How many different four-letter radio station call letters can be formed if the first letter must be W or K?6. A social security number contains nine digits. How many different ones can be formed?

7 and 8

7. If you wish to arrange your 7 favorite books on a shelf, how many different ways can this be done?8. If you have 10 favorite books, but only have room for 7 books on the shelf, how many ways can you arrange them?

9, 10 and 11

9. You have 20 favorite photographs and wish to arrange 12 of them on a mantel. How many ways can that be done?10. You take a multiple choice test with 12 questions (and each can be answered A B C D E). How many different ways could you answer the test?11. If 15 movies came out this month, how many ways could you pick your 1st, 2ndand 3rdfavorite?

12. How many ways can you rearrange the letters in

CAT?BOB?OHIO?CLASSES?e. MISSISSIPPI?

Another Counting Rule

Permutations of Duplicate itemsThe number of permutations of n items, where p items are identical, q items are identical, r items are identical, and so on, is given byExample:If a station plans on running 6 (identical) Democratic ads, 6 (identical) Republican ads, and 4 (identical) Independent ads, in how many ways can they order these?

Combination Questions

If there are 4 people in the math club (Anne,Bob,Cindy,Dave), and 2 will be selected to attend the national math conference. LIST all of the different ways that this is possible.From these 4 people (Anne,Bob,Cindy,Dave), and 3 will be selected to attend the national math conference. LIST all of the different ways that this is possible.

Permutationsand Combinations

PermutationsUse when ORDER matters and NO repetitionnPr= n!/(n-r)!Example: If 10 people join a club, how many ways could we pick pres andvp? 10P2 = 90CombinationsUse: ORDER does NOT matter and NO repetitionnCr= n!/ [(n-r)!r!]Example: 10 people join a club. In how many ways could we pick 2? 10C2 = 45

Combination of n things taken r at a time

Use the combination formulanCr= n!/[(n-r)!r!]:1. If there are 20 people on a committee, how many ways could we pick a subcommittee of 7 of them?2. If there are 72 potential jurors, how many different ways could they pick a jury of 12?

Decide and answer: Combination, permutation, or multiplication?

There are 8 possible pizza toppings. How many ways could we pick 3 toppings?2 . 20 people apply for a scholarship. 3 are chosen. In how many ways can they be chosen?

Changesome of the following permutation problems into combination problems

1. Permutation question: With 14 players on a team, how many ways could we pick a batting order of 11? Answer: 14P11Write a combination questions whose answer is 14C112. Permutation question: If you have 10 favorite books, but only have room for 7 books on the shelf, how many ways can you arrangethem?Answer: 10P7Write a combination questions whose answer is 10C7

More challengingproblems

1 If we have 4 teachers and 7 students and wish to form a committee of 2 teachers and 3 students, in how many different ways can this be done?2 . A test has 5 essay questions and 10 short answer questions. A student is to select to answer 3 essayand7 short answers. In how many different ways could this be done?

Review --Multiplication, Permutation, or Combination?

Recall:In permutations, does order matter?Is repetition allowed?In combinations, does order matter?Is repetition allowed?If order matters and repetition is allowed, what type of problem is this?1. With 13 players on a team, how many ways could we pick a batting order of 9?2. If a password has have 3 letters and then 5 numbers, how many ones exist?

Review3. If you have 10 favorite sweaters, but only have room for 4 in your suitcase, how many ways can you chose them?4. You have 18 favorite photographs and wish to arrange 11 of them in a line on your door. How many ways can that be done?5. You take a multiple choice test with 60 questions (and each can be answered A B C D ). How many different ways could you answer the test?6. If you had 11 soup ingredients, how many ways could you pick 5 of them?

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