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

Review of 5.1 SUM rule Product rule Inclusion/Exclusion Complement

Review of 5.3 Order matters, repetition allowed – Multiplication Rule – Ex: Social Security numbers 10 9 Order matters, repetition NOT allowed – Permutations: P(n,r)= – Ex: number of ways to pick 1 st, 2 nd, 3 rd from 30P(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 30C(30,3)=4060 Permutations of sets with indistinguishable objects – section 5.5: – Ex: number of ways to rearrange the letters in MISSISSIPPI (order matters)

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?

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

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?

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.

A few examples- two $10s, two $5s, one $1 $100$50$20$10$5$2$1 xx x

$100$50$20$10$5$2$1 xxxxx

$100$50$20$10$5$2$1

$100$50$20$10$5$2$1

$100$50$20$10$5$2$1

$100$50$20$10$5$2$1

$100$50$20$10$5$2$1

solution $100$50$20$10$5$2$1

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

more examples on #2, solution ChocChoc chipPbSugaroat

Ex #3: How many solutions does the equation x 1 +x 2 +x 3 +x 4 = 20 have where x 1, x 2, x 3, x 4 are nonnegative integers?

Solution

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

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?

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?

a) A dozen croissants PlainCherryChocAlmondAppleBroccoli

b) 3 dozen croissants PlainCherryChocAlmondAppleBroccoli

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

d) 2 dozen, with no more than 2 broccoli? PlainCherryChocAlmondAppleBroccoli

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

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? PlainCherryChocAlmondAppleBroccoli