Review of 5.1, 5.3 and new Section 5.5: Generalized Permutations and Combinations
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Transcript of Review of 5.1, 5.3 and new Section 5.5: Generalized Permutations and Combinations
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Review of 5.1, 5.3 and
new Section 5.5: Generalized Permutations and Combinations
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Review of 5.1
• SUM rule• Product rule• Inclusion/Exclusion• Complement
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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)
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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?
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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
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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?
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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.
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A few examples- two $10s, two $5s, one $1
$100 $50 $20 $10 $5 $2 $1
xx xx x
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$100 $50 $20 $10 $5 $2 $1
x x xx x
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• $100 $50 $20 $10 $5 $2 $1
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• $100 $50 $20 $10 $5 $2 $1
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•
$100 $50 $20 $10 $5 $2 $1
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•
$100 $50 $20 $10 $5 $2 $1
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$100 $50 $20 $10 $5 $2 $1
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solution
• $100 $50 $20 $10 $5 $2 $1
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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
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more examples on #2, solutionChoc Choc chip Pb Sugar oat
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Ex #3: How many solutions does the equation x1+x2+x3+x4 = 20 have where x1, x2, x3, x4 are nonnegative integers?
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Solution
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Review: Permutations of sets with indistinguishable objects
Ex. 4: How many ways can we rearrange the letters: BOB CLASSES ARKANSAS
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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?
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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?
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a) A dozen croissantsPlain Cherry Choc Almond Apple Broccoli
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b) 3 dozen croissants
Plain Cherry Choc Almond Apple Broccoli
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C) 2 dozen, with at least 2 of each kind?
Plain Cherry Choc Almond Apple Brocolli
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d) 2 dozen, with no more than 2 broccoli?
Plain Cherry Choc Almond Apple Broccoli
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e) 2 dozen, with at least 5 chocolate and at least 3 almond?
Plain Cherry Choc Almond Apple Broccoli
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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