So far…we have looked at:
1. Fundamental Counting Rule –
How many outcomes are possible with 5 rolls of a die?
2. Permutations –
How many distinct ways can you arrange the letters of the
Objective: Chapter 2 - Systematic Lists 1
We will continue to work with Permutations
and take a look at Combinations and throw in
a quick review of the Fundamental Counting
Objective: Chapter 2 - Systematic Lists 2
Quick Review of 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 times n ways.
Example: How many different combinations
of sandwiches are possible with a choice of 3
types of bread, 4 types of meat and 3 types
Objective: Chapter 2 - Systematic Lists 3
How many different combinations are
possible on a lock with 4 tumblers with the
digits 0-9 each on them?
Objective: Chapter 2 - Systematic Lists 4
A collection of n different items can be arranged in order n!
different ways (This factorial rule reflects the fact that the
first item may be selected n different ways, the second item
may be selected n-1 ways, and so on).
Example: You have just started your own airline company
called Air America. You have one plane for a route
connecting Austin, Boise, and Chicago. How many routes
Objective: Chapter 2 - Systematic Lists 5
How many different ways can a series of 5
questions be asked?
Objective: Chapter 2 - Systematic Lists 6
The number of permutations (or sequences) of r items
selected from n available items (without replacement):
(n - r)!
Example: We are going to conduct a survey. We would like
to visit every state capital to ask our poll questions. That
isn’t going to be possible. We are going to visit 4 capitals.
How many different routes are possible (example:
Sacramento, Salem, Phoenix, Boise is not the same as
Boise, Phoenix, Salem, Sacramento)
Objective: Chapter 2 - Systematic Lists 7
Permutations with Duplicate Items
If there are n items with n1 alike, n2 alike,…,nk alike, the
number of permutations of all n items is:
n1! . n2! .. . . . . . . nk!
Example: The letters DDDDRRRRR, represent a sequence
of diet and regular cola. How many ways can we arrange
Objective: Chapter 2 - Systematic Lists 8
You have 3 red marbles, 3 green marbles
and 4 blue marbles. How many different
ways can the marbles be arranged?
Objective: Chapter 2 - Systematic Lists 9
The number of combinations of r items selected
from n different items is:
nCr = (n - r )! r!
“n” different items
“r” items to be selected
different orders of the same items are not counted
Objective: Chapter 2 - Systematic Lists 10
In the New York State lottery, a player wins first prize by
selecting the correct 6-number combination when 6 different
numbers from 1 through 51 are drawn. If a player selects
one particular 6-number combination, find the probability of
winning (The player need not select the 6 numbers in the
same order as they are drawn, so order is irrelevant).
Objective: Chapter 2 - Systematic Lists 11
A standard deck of 52 playing cards has 4
suits with 13 different cards in each suit.
If the order in which the cards are dealt is not
important, how many different 5-card hands
Objective: Chapter 2 - Systematic Lists 12
Big Red Flag!!
When different orderings of the same items
are counted separately, we have a
permutation problem, but when different
orderings of the same items are not counted
separately, we have a combination problem.
Objective: Chapter 2 - Systematic Lists 13
Telling the Difference
Example: How many different ways can you
arrange the first four hitters on a baseball
team of 9?
Now, how many ways can you arrange the
first four hitters if order matters?
Objective: Chapter 2 - Systematic Lists 14
Let’s Try Some
Tell whether you would use a combination or a
1. The winner and first, second, and third
runner up in a contest with 10 finalists.
2. Selecting two of eight employees to attend
a business seminar.
3. An arrangement of the letters in the word
Objective: Chapter 2 - Systematic Lists 15
Counting Worksheet due Thursday.
Objective: Chapter 2 - Systematic Lists 16