Review of 5.1, 5.3 and new Section 5.5: Generalized Permutations and Combinations

1. Review of 5.1, 5.3 andnew Section 5.5: Generalized Permutations and Combinations

2. Review of 5.1 • SUM rule • Product rule • Inclusion/Exclusion • Complement

3. Review of 5.3 • Order matters, repetition allowed • Multiplication Rule • Ex: Social Security numbers 109 • Order matters, repetition NOT allowed • Permutations: P(n,r)= • Ex: number of ways to pick 1st, 2nd, 3rd from 30 P(30,3)=30*29*28=24,360 • Order DOESN’T matter, repetition allowed • section 5.5: Combinations with Repetition: C(n+r-1,r)= • Ex: number of ways to pick several types of donuts, with more than 1 of each kind (order doesn’t matter) • Order DOESN’T matter, repetition NOT allowed • Combinations: C(n,r)= • Ex: number of ways to pick a committee of 3 from 30 C(30,3)=4060 • Permutations of sets with indistinguishable objects • section 5.5: • Ex: number of ways to rearrange the letters in MISSISSIPPI (order matters)

4. 5.3 review problems #1) If 4 people out of 35 are selected to win a $10 gift certificate, how many ways could they be chosen? #2) How many subsets of {a,b,c,d} exist? #3) 15 women and 7 men show up for jury duty. How many ways could you pick 8 women and 4 men?

5. More 5.3 examples • #4) How many bit strings of length 10 have: • Exactly three 0’s • The same number of 0s and 1s • At least seven 1s • At least two 1s

6. More 5.3 Examples • #5: If you make passwords out of either digits or letters, how many • 8 character passwords exist? • With no digits • With one digit • With at least one digit • With two digits • With at least 2 digits?

7. New Material– Section 5.5:Ex. 1(example 3 in the book: p.372) • How many ways are there to select 5 bills from a money bag containing $1, $2, $5, $10, $20, $50, and $100 bills? Assume order does not matter and bills of each denomination are indistinguishable.

8. A few examples- two $10s, two $5s, one $1

15. solution

16. Ex. #2: Cookies- suppose a shop has 5 types of cookies. How many different way can we pick 7 cookies?

17. more examples on #2, solution

18. Ex #3: How many solutions does the equation x1+x2+x3+x4 = 20 have where x1, x2, x3, x4 are nonnegative integers?

19. Solution

20. Review: Permutations of sets with indistinguishable objects Ex. 4: How many ways can we rearrange the letters: BOB CLASSES ARKANSAS

21. More examples • How many ways could a radio announcer decide the order that 6 (identical) Republican ads, 5 Democrats ads, and 4 Independent ads will play?

22. Ex #5: Donuts Ex 5: A croissant shop has plain, cherry, chocolate, almond, apple, and broccoli croissants (6 types). How many ways are there to choose: a) a dozen croissants b) 3 dozen croissants c) 2 dozen, with at least 2 of each kind? d) 2 dozen, with no more than 2 broccoli? e) 2 dozen, with at least 5 chocolate and at least 3 almond? f) 2 dozen, with at least 1 plain, at least 2 cherry, at least 3 chocolate, at least 1 almond, at least 2 apple, and no more than 3 broccoli?

23. a) A dozen croissants

24. b) 3 dozen croissants

25. C) 2 dozen, with at least 2 of each kind?

26. d) 2 dozen, with no more than 2 broccoli?

27. e) 2 dozen, with at least 5 chocolate and at least 3 almond?

28. f) 2 dozen, with at least 1 plain, at least 2 cherry, at least 3 chocolate, at least 1 almond, at least 2 apple, and no more than 3 broccoli?