ELEMENTARY
Section 3-6
STATISTICS
Counting
MARIO F. TRIOLA
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
EIGHTH
EDITION
1
Assume a quiz consists of two
questions. A true/false and a
multiple choice with 5 possible
answers. How many different ways
can they occur together.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
2
Tree Diagram of the events
Assume a quiz consists of two questions. A true/false and a
multiple choice with 5 possible answers. How many
different ways can they occur together.
a
T&a
b
T&b
c
T&c
T
d
T&d
e
T&e
a
F&a
b
F&b
c
F
F&c
d
F&d
e
F&e
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
3
Tree Diagram of the events
Let m represent the number of ways the first event can occur.
Let n represent the number of ways the second event can occur.
a
T&a
b
T&b
c
T&c
T
d
T&d
e
T&e
a
F&a
b
F&b
c
F
F&c
d
F&d
e
F&e
m=2
n=5
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
m*n = 10
4
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, together the events can
occur a total of m • n ways.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
5
Example
An ATM pin number is a 4 digit number. How
many possible pin numbers are there, if you
allow repeats in each position?
Digit:
# of Choices:
1st
2nd
3rd
4th
10
10
10
10
By the FCR, the total number of possible outcomes are:
10 * 10 * 10 * 10 = 10,000
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
6
Example
An ATM pin number is made up of 4 digit
number. How many possible outcomes are
there, if no repeats are allowed?
Digit:
# of Choices:
1st
10
2nd
3rd
4th
9
8
7
By the FCR, the total number of possible outcomes are:
10 * 9 * 8 * 7 = 5040
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
7
Example
Rank three players (A, B, C). How many
possible outcomes are there?
Ranking: First
Number of Choices:
Second
Third
2
1
3
By FCR, the total number of possible outcomes are:
3*2*1=6
( Notation: 3! = 3*2*1 )
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
8
Notation
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
The ! key on your TI-8x calculator is found by
pressing MATH and selecting PRB and selecting
choice #4 !
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
9
Factorial Rule
An entire collection of n different items can
be arranged in order n! different ways.
Example:
How many different seating charts could be
made for a class of 13?
13! = 6,227,020,800
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
10
Example
Eight players are in a competition, three of
them will win prizes (gold/silver/bronze).
How many possible outcomes are there?
Prizes:
Number of Choices:
gold
silver
bronze
8
7
6
By FCR, the total number of possible outcomes are:
8 * 7 * 6 = 336 = 8! / 5!
= 8!/(8-3)!
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
11
Permutations Rule
 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
P
n r =
n!
(n – r)!
 Order is taken into account
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
12
Permutations Rule
when some items are identical to
others
If there are n items with n1 alike, n2 alike, . . .
nk alike, the number of permutations is
n!
n1! . n2! .. . . . . . . nk!
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
13
Permutations Rule
How many ways can the letters in
MISSISSIPPI be arranged?
I occurs 4 times
S occurs 4 times
P occurs 2 times
11!
=34,650
4 !4 !2 !
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
14
Example
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)! = 336
 For each chosen top three, if we rank/order them,
there are 3! possibilities.
==> the number of choices of Top 3 without order
8!
are {8!/(8-3)!}/(3!) =
= 56
(8-3)! 3!
Combinations rule
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
15
Combinations Rule
 the number of combinations is
n!
nCr =
(n – r )! r!
 n different items
 r items to be selected
 different orders of the same items are
not counted
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
16
Permutation –Order Matters
Combination - Order does not matter
TI-83/4
Press MATH choose PRB choose 2: nPr or 3:
nCr to compute the # of outcomes.
Example:
10P5
= 10 nPr 5 = 30240
10C5
= 10 nCr 5 = 252
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
17
Counting Devices Summary
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
18
Counting Devices Summary
Are there n different items with all of them
to be used in different arrangements?
If so, use the factorial rule and find n!.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
19
Counting Devices Summary
Are there n different items with some of
them to be used in different arrangements?
If so, evaluate
n!
n Pr =
(n – r )!
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
20
Counting Devices Summary
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
n!
n1! n2!. . . . . . nk!
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
21
Counting Devices Summary
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
n!
nCr =
(n – r )! r!
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
22