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Review of 5.1, 5.3 andnew Section 5.5: Generalized Permutations and Combinations Review of 5.1SUM ruleProduct ruleInclusion/ExclusionComplementReview of 5.3Order matters, repetition allowedMultiplication RuleEx: Social Security numbers109Order matters, repetition NOT allowedPermutations: P(n,r)= Ex: number of ways to pick 1st, 2nd, 3rd from 30P(30,3)=30*29*28=24,360Order DOESNT matter, repetition allowedsection 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 doesnt matter)Order DOESNT matter, repetition NOT allowedCombinations: C(n,r)= Ex: number of ways to pick a committee of 3 from 30C(30,3)=4060Permutations of sets with indistinguishable objectssection 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 0sThe same number of 0s and 1sAt least seven 1sAt least two 1sMore 5.3 Examples#5: If you make passwords out of either digits or letters, how many8 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$1xxxxx $100$50$20$10$5$2$1xxxxx $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$1Ex. #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, solutionChoc Choc chipPbSugaroatEx #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: BOBCLASSESARKANSAS

More examplesHow 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: DonutsEx 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 croissantsPlainCherryChocAlmondAppleBroccoli

b) 3 dozen croissants

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

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

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

PlainCherryChocAlmondAppleBroccolif) 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