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Lesson 9: Combinatorix
Here we begin methods of counting
First we need factorial notation:
n
! =
n
(
n

1)(
n

2)
· · ·
2
·
1
Examples: 0! = 1, 1! = 1, 3! = 3
*
2
*
1 = 6, 6! = 6
*
5
*
4
*
3
*
2
*
1 = 720
Counting with order
Lets say we have 5 books. How many ways can we put them all on the
bookshelf?
In the ±rst spot, there are 5 books we can pick. In the second spot there
are 4 books left. In the third spot, there are 3 possibilities. In the fourth
spot, there are 2 books to choose from. Finally, there is only one book left in
the last spot. So we get 5! ways to order the books.
So for n books, there are
n
! ways to order them without repetition.
If we have
m
books and
n
spots on the bookshelf, we get
m
!
/
(
m

n
)! ways
to order them without repetition or
m
n
ways to order them with repetition.
Try it for 5 books and 3 spots.
Counting without order
What if we don’t care about order.
Lets say we have 5 books. How many ways can we pick 3 to bring to
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This note was uploaded on 03/22/2010 for the course MATH 1104 taught by Professor Unknown during the Fall '10 term at Carleton.
 Fall '10
 Unknown
 Linear Algebra, Algebra, Counting

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