104- Ch. 11 part 1; 260- Ch. 5 pt. 2 master_prob_pc.pptx Probability Theory-- Counting: Multiplication, Permutations, Combinations 104- Ch. 11 part 1; 260- Ch. 5 pt. 2 master_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 DA AC BC CB DB AD BD CD DC 4*3=12 or 4P2 = 12

4*3*2 = 24 outcomes Or 4P3 = 24 A B C ABC D ABD C B ACB D ACD D A BDA C BDC B A C BAC D BCD C A BCA C A B CAB D CAD B A CBA D CBD A B DAB C DAC D A B DAB B A DBA C DBC C A DCA B DCB 4*3*2 = 24 outcomes Or 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. Multiple choice test A teacher 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 use a 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 Rule             If 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 time             nPr =     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?

…decide which 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 , 2nd and 3rd favorite?

12. How many ways can you rearrange the letters in   CAT? BOB? OHIO? CLASSES?  e.      MISSISSIPPI?     

Another Counting Rule Permutations of Duplicate items             The number of permutations of n items, where p items are identical, q items are identical, r items are identical, and so on, is given by Example: 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.

Permutations and Combinations Use when ORDER matters and NO repetition nPr = n!/(n-r)! Example: If 10 people join a club, how many ways could we pick pres and vp? 10P2 = 90 Combinations Use: ORDER does NOT matter and NO repetition nCr = 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 formula nCr = 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?

Change some 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: 14P11 Write a combination questions whose answer is 14C11 2. Permutation question: If you have 10 favorite books, but only have room for 7 books on the shelf, how many ways can you arrange them?Answer: 10P7 Write a combination questions whose answer is 10C7

More challenging problems 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 essay and 7 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? 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?

Review 3. 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?