MATH370
Section 5.5: Generalized Permutations and Combinations
Review of different types of problems:
•
•
•
•
•
Order matters, repetition allowed
–
Multiplication Rule
–
Ex: Social Security numbers
109
Order matters, repetition NOT allowed
𝑛!
–
Permutations: P(n,r)= (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
(𝑛+𝑟−1)!
–
section 5.5: Combinations with Repetition:
–
Ex: number of ways to pick several types of donuts, with more than 1 of each kind
(order doesn’t matter)
C(n+r-1,r)= 𝑟!(𝑛−1)!
Order DOESN’T matter, repetition NOT allowed
𝑛!
–
Combinations: C(n,r)= (𝑛−𝑟)!𝑟!
–
Ex: number of ways to pick a committee of 3 from 30
C(30,3)=4060
Permutations of sets with indistinguishable objects
𝑛!
–
section 5.5:
–
Ex: number of ways to rearrange the letters in MISSISSIPPI (order matters)
𝑛1 !∗𝑛2 !∗…∗𝑛𝑘 !
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.
First a few examples
$100
$50
|
$100
X
X
$20
|
$50
X
|
x
$10
Xx
|xx
$20
|
$10
Xx
|xx
$5
Xx
$2
|xx
|
$5
|
$2
X
$1
x
|x
$1
|x
|
$100
$50
$20
$10
$5
$2
$1
$100
$50
$20
$10
$5
$2
$1
n=7, n-1=6, r=5
C(n-1+r,r)=C(11,5) or C(11,6)
Ex. 2: Suppose a shop has 5 types of cookies. How many different ways can we pick 7 cookies?
Ex:
Choc
choc-chip
pb
sugar
oat
Xx
|xx
|x
|x
|x
N=5, n-1=4, r=7
C(11,7)=
Ex. 3: How many solutions does the equation x1+x2+x3+x4 = 20 have where x1, x2, x3, x4 are nonnegative
integers?
Ex:
5+5+5+5
xxxxx|xxxxx|xxxxx|xxxxx
4+5+5+6
xxxx|xxxxx|xxxxx|xxxxxx
10+8+1+1
1+1+8+10
0+5+0+15
C(23,20)=
Ex. 4: How many ways can we rearrange the letters:
BOB
OHIO
CLASSES
MISSISSIPPI
Ex 5: A croissant shop has plain, cherry, chocolate, almond, apple, and broccoli croissants (6 types). How
many ways are there to choose:
a)
b)
c)
d)
e)
f)
a dozen croissants
3 dozen croissants
2 dozen, with at least 2 of each kind?
2 dozen, with no more than 2 broccoli?
2 dozen, with at least 5 chocolate and at least 3 almond?
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?
Answers:
a)
b)
c)
d)
e)
f)
17C5
41C5
12 so far… 17C5
Total minus 3 or more broccoli (21 left) ….29C5 -26C5
16 left… 21C5
15 left – 11 left… 20C5 – 16?C 11