MATH260Ch. 5: Probability Theory

part 4

Counting: Multiplication, Permutations, Combinations

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 DC

4*3=12 or 4P2 = 12

Answers2. 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.

ABCABD…

• A B C ABC

D ABDC B ACB

D ACDD A BDA

C BDC• B A C BAC

D BCDC A BCA

D BCDD A BDA

C BDC• C A B CAB

D CADB A CBA

D CBDA B DAB

C DAC• D A B DAB

C DACB A DBA

C DBCC A DCA

B DCB

4*3*2 = 24 outcomesOr 4P3 = 24

More counting examples:

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

2. T/F quiz

2. 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. LIST them all.

3. T/F test

3. What if there were 10 T/F questions. Just explain (do not list).

4. Multiple choice test

4. 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). LIST them all.

5. And 6.5. What if there were 20 multiple choice

questions with 5 choices each? Explain (don’t list).

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

Counting Rules Fundamental Counting/ –Multiplication Rule (p. 608) 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 (p. 617) nPr = n!/(n-r)! In permutations, ORDER matters & REPETITION is NOT allowed? Permutations of Duplicate items (p. 618) 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

More multiplication and permutation problems

1. With 14 players on a team, how many ways could we pick a batting order of 11?

2. If license plates have 3 letters and then 4

numbers, how many different license plates exist?

3

3. A stock can go up, down, or stay unchanged. If you own 7 stocks, how many different possibilities are there?

4

4. How many different four-letter radio station call letters can be formed if the first letter must be W or K?

5. A social security number contains nine

digits. How many different ones can be formed?

6

6. If you wish to arrange your 7 favorite books on a shelf, how many different ways can this be done?

7. If you have 10 favorite books, but only have

room for 7 books on the shelf, how many ways can you arrange them?

8

8. You wish to arrange 12 of your favorite photographs on a mantel. How many ways can this be done?

9. You have 20 favorite photographs and wish

to arrange 12 of them on a mantel. How many ways can that be done?

10.

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. How many ways can you rearrange the letters in

a. CAT?

b. OHIO?

c. CLASSES?

d. MISSISSIPPI?

12

12. 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?

13. If you saw 15 movies last year, how many

ways can the top 3 be chosen and ranked?

14

14. 20 people purchase raffle tickets. How many ways could we award a 1st, 2nd, and 3rd prize.

15. You have 50 different outfits. How many

ways can you pick your first and second favorite? How about your first, second, and third favorite?

Combination Questions

1. 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.

2. 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.

Combination answers1. 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.

ABAC BCAD BD CD

4C2= 6

Combination answer

2. 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.

ABC BCDABDACD

4C3 = 4

Permutations and Combinations• Permutations– 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 (p. 623)

Use the combination formula nCr = n!/[(n-r)!r!] to answer these combination problems

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 100 senators, how many ways could we pick a subcommittee of 7 of them?

3 If there are 72 potential jurors, how many different ways could they pick a jury of 12?

Decide and answer: Combination, permutation, or multiplication?

1. 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?

3. 32 people are in a class where the teacher plans on awarding 4 A’s. If all possibilities were written out, how many would there be?

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

…3. Permutation question: You have 20 favorite photographs

and wish to arrange 12 of them on a mantel. How many ways can that be done? Answer: 20P12

Write a combination questions whose answer is 20C12 4. Permutation question: If you saw 15 movies last year, how

many ways can the top 3 be chosen and ranked? Answer: 15P3

Write a combination questions whose answer is 15C3

5. Permutation question: 20 people purchase raffle tickets. How many ways could we award a 1st, 2nd, and 3rd prize. Answer: 20P3

Write a combination questions whose answer is 20C3

More challenging combination 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 questions and 7 short answers. In how many different ways could this be done?

Review --Multiplication, Permutation, or Combination?

1. With 14 players on a team, how many ways could we pick a batting order of 11?

2. If license plates have 3 letters and then 4 numbers, how many different

license plates exist? 3. How many different four-letter radio station call letters can be formed if

the first letter must be W or K? 4. A social security number contains nine digits. How many different ones

can be formed? 5. If you wish to arrange your 7 favorite books on a shelf, how many

different ways can this be done?

6. If you have 10 favorite books, but only have room for 7 books on the shelf, how many ways can you arrange them?

7. You wish to arrange 12 of your favorite photographs on a mantel. How many ways can this be done?

8. You have 20 favorite photographs and wish to arrange 12 of them on a mantel. How many ways can that be done?

9. 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?

10. If you had 13 pizza toppings, how many ways could you pick 5 of them?