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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 examplestwo $10s, two $5s, one $1 $100 $50 $20 $10 $5 xx xx $2 $1 x $100 $50 x x $20 $10 xx $5 $2 x $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 $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, solution Choc 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 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 Plain 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