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