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Definition of the Factorial
We define n! recursively by
0! = 0, 1! = 1,
n! = n(n - 1)!
Example:
5! = 5(4)(3)(2) = 120
Example:
Suppose that we are interested in how many ways there are in scrambling the
letters of the name "Cindy". We have 5 choices for the first letter,
once we have chosen the first letter there are 4 choices for the second letter,
and then three choices for the third letter, two for the fourth letter, and
only one choice for the last letter. Hence there are
5(4)(3)(2)(1) = 5!
choices.
Permutations
Example
If we want to select only three letters from the word "Cindy" then we have
(5)(4)(3) = 5!/(5 -
3)!
choices.
Definition
The number of permutations of n distinct objects
taken r at a time is
nPr = n!/(n - r)! |
You can find this button on the TI 85 calculator by hitting Math -> Prob
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Distinguishable Permutations
Example
How many ways are there of scrambling the name Tamara Heether?
Solution:
If there were no duplicate letters the solution would
be 13!, but this is not the case. There are
2 T's, 3 A's
2 R's and
3E's
We must divide by 2!3!2!3! to get
13!/[2! 3! 2! 3!] = 43,243,200
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Theorem
If there are n objects with n1 duplicates
of one kind, n2 duplicates of a second kind, ...,
nk
duplicates of a kth kind, then the number of distinguishable permutations
of these n objects is
n!/(n1!n2!...nk!) |
Exercise:
How many ways are there to scramble your first and
last name?
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Combinations
Example
How many different five card poker hands are there?
Solution
First note that there are 52P5 different ordered five card poker hands,
however, two hands that have the same five cards, but in a different order should not
be counted as distinct hands. Since there are 5! ways of ordering five
cards, we have
52P5/5! = 52!/[5!(52 - 5)!] = 2,598,960
different poker hands.
Note that only four of these hands are Royal Flushes, hence there is a 4
in 2,598,960 or about one in half a million chance of receiving a Royal
Flush in a 5 card stud poker game.
Theorem
The number of ways of choosing r objects from
n where
order does not matter is
nCr
= n!/(n - r)!r! |
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The Binomial Theorem
consider
(x + y)5 = (x + y)(x + y)(x + y)(x + y)(x + y)
Q: How many ways are there to select all x's?
A: 1 way.
Q: How many ways are there to select 4x's from the 5 possible?
A: 5C4
ways
Exercise:
How many ways are there to select two x's from the five?
These
investigations lead us to believe that
(x + y)5 = 5C5 x5 +
5C4
x4y + 5C3 x3y2 + 5C2
x2y3 + 5C1 xy4 5C0 y5
|
Theorem
(x + y)n = Si = 0n
nCn - i
xn-iyi
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Example
Find
(3x - 2y)4
Solution
The formula gives us
5C4 (3x)4 +
5C3 (3x)3(-2y) + 5C2
(3 x)2(-2y)2 + 5C1 (3 x)(-2y)3
5C0 (-2y)4
= 5(34x4)
+ 10(27x3)(-2y) + 10(9x2)(4y2) + 5(3x)(-8y3)
+ (16y4)
= 405x4 - 540x3y
+ 360x2y2 - 120xy3 + 16y4