copyright © 1998, triola, elementary statistics addison wesley longman 1 counting section 3-7 m a r...
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman1
CountingCountingSection 3-7Section 3-7
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman2
Fundamental Counting Rule
For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m • n ways.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman3
a
b
c
d
e
a
b
c
d
e
T
F
Tree Diagram of the events
T & aT & bT & cT & dT & eF & aF & bF & cF & dF & e
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman4
a
b
c
d
e
a
b
c
d
e
T
F
Tree Diagram of the events
T & aT & bT & cT & dT & eF & aF & bF & cF & dF & e
m = 2 n = 5 m*n = 10
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Rank three players (A, B, C). How many possible outcomes are there?
Ranking: First Second Third
Number of Choices: 3 2 1
By FCR, the total number of possible outcomes are:
3 * 2 * 1 = 6
( Notation: 3! = 3*2*1 )
Example
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The factorial symbol ! denotes the product of decreasing positive whole numbers.
n! = n (n – 1) (n – 2) (n – 3) • • • • • (3) (2) (1)
Special Definition: 0! = 1
Find the ! key on your calculator
Notation
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A collection of n different items can be arranged in order n! different ways.
Factorial Rule
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Eight players are in a competition, three of them will win prices (gold/silver/bronze). How many possible outcomes are there?
Prices: gold silver bronze
Number of Choices: 8 7 6
By FLR, the total number of possible outcomes are:
8 * 7 * 6 = 336 = 8! / 5!
= 8!/(8-3)!
Example
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Addison Wesley Longman9
n is the number of available items (none identical to each other)
r is the number of items to be selected
the number of permutations (or sequences) is
Permutations Rule
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Addison Wesley Longman10
n is the number of available items (none identical to each other)
r is the number of items to be selected
the number of permutations (or sequences) is
Permutations Rule
Order is taken into account
Pn r = (n – r)!n!
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when some items are identical to others
Permutations Rule
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when some items are identical to others
If there are n items with n1 alike, n2 alike, . . .
nk alike, the number of permutations is
Permutations Rule
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Addison Wesley Longman13
when some items are identical to others
If there are n items with n1 alike, n2 alike, . . .
nk alike, the number of permutations is
Permutations Rule
n1! . n2! .. . . . . . . nk! n!
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Eight players are in a competition, top three will be selected for the next round (order does not matter). How many possible choices are there?
By the Permutations Rule, the number of choices of Top 3 with order are 8!/(8-3)!
For each chosen top three, if we rank/order them, there are 3! possibilities.
==> the number of choices of Top 3 without order are {8!/(8-3)!}/(3!)
Example
8!
(8-3)! 3!=
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Addison Wesley Longman15
Eight players are in a competition, top three will be selected for the next round (order does not matter). How many possible choices are there?
By the Permutations Rule, the number of choices of Top 3 with order are 8!/(8-3)!
For each chosen top three, if we rank/order them, there are 3! possibilities.
==> the number of choices of Top 3 without order are {8!/(8-3)!}/(3!)
Example
8!
(8-3)! 3!= Combinations
rule!
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the number of combinations is
Combinations Rule
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n different items
r items to be selected
different orders of the same items are not counted
the number of combinations is
(n – r )! r!n!
nCr =
Combinations Rule
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When different orderings of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem.
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Is there a sequence of events in which the first can occur m ways, the second can
occur n ways, and so on?
If so use the fundamental counting rule and multiply m, n, and so on.
Counting Devices Summary
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Are there n different items with all of them to be used in different arrangements?
If so, use the factorial rule and find n!.
Counting Devices Summary
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Are there n different items with some of them to be used in different arrangements?
Counting Devices Summary
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Are there n different items with some of them to be used in different arrangements?
If so, evaluate
Counting Devices Summary
(n – r )!n!
n Pr =
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Are there n items with some of them identical to each other, and there is a need to find the total number of different arrangements of all of those n items?
Counting Devices Summary
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Are there n items with some of them identical to each other, and there is a need to find the total number of
different arrangements of all of those n items?
If so, use the following expression, in which n1 of the
items are alike, n2 are alike and so on
Counting Devices Summary
n!n1! n2!. . . . . . nk!
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Addison Wesley Longman25
Are there n different items with some of them to be selected, and is there a need to find the total number of combinations (that is, is the order irrelevant)?
Counting Devices Summary
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman26
Are there n different items with some of them to be selected, and is there a need to find the total number of combinations (that is, is the order irrelevant)?
If so, evaluate
Counting Devices Summary
n!nCr = (n – r )! r!
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