Counting Principles
The Fundamental Counting Principle:
If one event can occur m ways and another can occur n ways, then the number of ways the events can occur in sequence is m*n.
********************************************Example:
A die roll can result in six different outcomes: 1,2,3,4,5,6. A coin flip can result in 2 different outcomes: H or T
A die roll and a coin flip can result in 2*6 = 12 different outcomes: 1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T
Pictures: http://commons.wikimedia.org/
Counting Principles
Example:
License Plates have 3 digits followed by 3 letters. How many license plates are
possible?
__ __ __ __ __ __
Counting Principles
Example:
License Plates have 3 digits followed by 3 letters. How many license plates are
possible?
__ __ __ __ __ __
10*10*10*26*26*26 = 17,576,000
Counting Principles
Example:
You have 2 pairs of socks, 3 pairs of pants and 2 shirts. How many different outfits
can you make?
___ ___ ___ socks shirts pants
Counting Principles
Example:
You have 2 pairs of socks, 3 pairs of pants and 2 shirts. How many different outfits
can you make?
_2_*_3_*_2_ = 12 socks pants shirts
Counting Principles
Factorials
N! = n*(n-1)*(n-2)…1
Examples:5! = 5*4*3*2*1 = 120
7! = 7*6*5*4*3*2*1 =5040
Counting Principles
Permutation: An ordered arrangement of objects (no repetition and order matters)
)!(
!
rn
nPrn
601*2
1*2*3*4*5
!2
!5
)!35(
!535
P
Example:
Counting Principles
Permutation: An ordered arrangement of objects (no repetition and order matters)
Another way to look at it: Three slots and five objects to choose from to fill them without replacement:
___*___*___35P
Counting Principles
Permutation: An ordered arrangement of objects (no repetition and order matters)
_5_ _4_ _3_= 6035P
Example:
Counting Principles
Permutation: An ordered arrangement of objects (no repetition and order matters)
How many ways can five people finish a race 1st, 2nd and 3rd?
_5_*_4_*_3_= 60 first second third
35P
Example:
Counting Principles
Combination: selection of r objects from a group of n objects (no repetition and order
does not matter)
Notice that this is the same formula as for a permutation, but you divide by r! because order does not matter and the objects can be ordered in r! ways
)!(!
!
rnr
nCrn
101*2*1*2*3
1*2*3*4*5
!2!3
!5
)!35(!3
!535
C
Counting Principles
Combination: selection of r objects from a group of n objects (no repetition and order
does not matter)
)!(!
!
rnr
nCrn
Notice that this is the same formula as for a permutation, but you divide by r! because order does not matter and the objects can be ordered in r! ways
!r
PC rnrn
Counting Principles
Combination: selection of r objects from a group of n objects (no repetition and order
does not matter)
________________ = 60/6 = 10
3!35C
5 4 3
Counting Principles
Combination: selection of r objects from a group of n objects (no repetition and order
does not matter)
How many ways are there to choose a three member team from five people?
________________ = 60/6 = 10
3!
35C5 4 3
Divide by the number of ways to order three objects
Counting Principles
Distinguishable PermutationsIf there are n1 of one type of object and n2 of another
type and there are n total, then there are
distinguishable ways of arranging them.
Example: How many distinguishable ways can you arrange AAABB?
!!
!
21 nn
n
102
20
)1*2(*)1*2*3(
1*2*3*4*5
!2!3
!5
Counting Principles
Distinguishable Permutations
Example: How many distinguishable ways can you arrange the letters in Mississippi?
34650!2!4!4
!11
Probability
We can apply these rules to probability:
outcomes
successxP
#
#)(
How many ways can you be dealt a five diamond hand from a deck of cards?
We choose five cards from the 13 diamonds, then divide by the number of ways to choose five cards from all 52
0005.)(552
513 C
CxP
Probability
We can apply these rules to probability:
outcomes
successxP
#
#)(
How many ways can you be dealt any flush from a deck of cards?First choose the suite from 4 suites, then choose five cards from 13 of that suite:
002.*
)(552
51314 C
CCxP
Probability
We can apply these rules to probability:
outcomes
successxP
#
#)(
How many ways can you be dealt a full house from a deck of cards?First choose the card for the three of a kind. Then, choose 3 of those cards, then choose the card for the two of a kind, then choose two of those cards:
001.***
)(552
2411234113 C
CCCCxP
Probability
Another example:
You have 25 students in a class. 20 are passing. You choose5 students. What is the probability you choose three passingstudents and two failing?
A distribution called the hypergeometric distribution is based on these kind of situations. We won’t worry about that now.
20 5
3 2.21
25
5
Top Related