# Combinations

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```					                                 “Counting” Continued…

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
word Mississippi?

Objective: Chapter 2 - Systematic Lists                     1
Today…

We will continue to work with Permutations
and take a look at Combinations and throw in
a quick review of the Fundamental Counting
Rule.

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
of cheese?

Objective: Chapter 2 - Systematic Lists                   3
You Try

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
Factorial Rule

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
are possible?

Objective: Chapter 2 - Systematic Lists                          5
You Try

How many different ways can a series of 5

Objective: Chapter 2 - Systematic Lists             6
Permutations Rule

The number of permutations (or sequences) of r items
selected from n available items (without replacement):
n!
nPr         =
(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:

n!
n1! . n2! .. . . . . . . nk!
Example: The letters DDDDRRRRR, represent a sequence
of diet and regular cola. How many ways can we arrange
those letters/colas?

Objective: Chapter 2 - Systematic Lists                            8
You Try

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
Combinations

The number of combinations of r items selected
from n different items is:

n!
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
Example

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
You Try

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
are possible?

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
permutation:
1. The winner and first, second, and third
runner up in a contest with 10 finalists.
2. Selecting two of eight employees to attend
3. An arrangement of the letters in the word
algebra?

Objective: Chapter 2 - Systematic Lists                    15
Homework

Counting Worksheet due Thursday.

Objective: Chapter 2 - Systematic Lists              16

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