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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 ictures: http://commons.wikimedia.org/

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Page 1: 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

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/

Page 2: 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

Counting Principles

Example:

License Plates have 3 digits followed by 3 letters. How many license plates are

possible?

__ __ __ __ __ __

Page 3: 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

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

Page 4: 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

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

Page 5: 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

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

Page 6: 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

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

Page 7: 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

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:

Page 8: 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

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

Page 9: 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

Counting Principles

Permutation: An ordered arrangement of objects (no repetition and order matters)

_5_ _4_ _3_= 6035P

Example:

Page 10: 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

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:

Page 11: 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

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

Page 12: 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

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

Page 13: 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

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

Page 14: 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

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

Page 15: 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

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

Page 16: 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

Counting Principles

Distinguishable Permutations

Example: How many distinguishable ways can you arrange the letters in Mississippi?

34650!2!4!4

!11

Page 17: 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

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

Page 18: 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

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

Page 19: 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

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

Page 20: 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

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


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