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 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)

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 xx x

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

x x xx x

• $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?

Chocolate Choc chip Pb Sugar oat

more examples on #2, solutionChoc Choc chip Pb Sugar oat

Ex #3: How many solutions does the equation x1+x2+x3+x4 = 20 have where x1, x2, x3, x4 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 croissantsb) 3 dozen croissantsc) 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 croissantsPlain Cherry Choc Almond Apple Broccoli

b) 3 dozen croissants

Plain Cherry Choc Almond Apple Broccoli

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

Plain Cherry Choc Almond Apple Brocolli

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

Plain Cherry Choc Almond Apple Broccoli

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

Plain Cherry Choc Almond Apple Broccoli

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?

Plain Cherry Choc Almond Apple Broccoli