Counting and Probability
It’s the luck of the roll.
Factorial
3! = 3 x 2 x 1 = 6
5! = 5 x 4 x 3 x 2 x 1= 120
X! = x(x-1)(x-2)…1
3! Is read as “3 factorial”
• On the TI-83 the factorial key is located by
pressing
MATH arrow over to PROB down to 4:!
Arrangements
• When we put things in order it is important
to know if repetition is allowed.
How many 3 digit codes can be made using
the numbers 0 – 9 if repetition is allowed?
If repetition is not allowed?
Examples
How many license plates can be made using 3
digits and 3 letters, repetition is allowed?
How many license plates can be made using 3
digits and 3 letters if the numbers cannot
repeat?
How many 7 character passwords can be made
using letters and numbers, repetition is allowed?
Fundamental Counting Principal
Assignment
You are the ruler of a small kingdom of
50,000 peasants, each owning a vehicle.
Your ego urges you to design a new
license plate showing your face as the
background. Using numbers, letters,
and/or symbols, what would be on your
license plate so that there will be enough
for every citizen and allow for 8% growth
during your 4-year rule?
Menu Madness
Using the menu from a local eatery, create
combination offerings that could be “red
plate specials.” From what items would
you have them choose and what would be
the cost?
List the possible selections for your specials.
Permutations
A permutation is an ordered arrangement.
The number of permutations for n objects is n!
n! = n (n – 1) (n – 2)…..3 • 2 • 1
The number of permutations of n
objects taken r at a time (r  n) is:
Example
You are required to read 5 books from a list of 8.
In how many different orders can you do so?
There are 6720 permutations of 8 books reading 5.
Combinations
A combination is a selection of r objects from a
group of n objects.
The number of combinations
of n objects taken r at a time is
Example
You are required to read 5 books from a list of 8.
In how many different ways can you choose the
books if order does not matter.
There are 56 combinations of 8 objects taking 5.
1
2
3
4
Combinations of 4 objects choosing 2
1
2
3
1
1
4
2
3
3
4
2
4
Each of the 6 groups represents a combination.
1
2
4
3
Permutations of 4 objects choosing 2
1
1
1
2
3
2
3
4
4
1
1
1
2
3
3
4
3
4
2
3
2
4
4
2
Each of the 12 groups represents a permutation.
Example
In a race with eight horses, how many
ways can three of the horses finish in
first, second and third place? Assume
that there are no ties.
Example
The board of directors for a company has
twelve members. One member is the
president, another is the vice president,
another is the secretary and another is
the treasurer. How many ways can
these positions be assigned?
More Menu Madness
Panda Express offers 3 entrees plus a side
for a given price. What will you offer?
Rather than have your eatery’s special
require people to pick from different
categories, you’re allowing them to pick
any 3 things from any where on the menu.
What might the cost be? How would you
write this selection option?
Distinguishable Permutations
The number of distinguishable
permutations of n objects where n1 are
one type and n2 are of another type and
so on, is:
n!
n1!•n2!•n3!•… nk!
Example
To find the number of permutations in the
word Mississippi
total letters
each set of repeats
=
11!
2!4!4!
Example
A contractor wants to plant six oak trees,
nine maple trees, and five poplar trees
along a subdivision street. If the trees
are spaced evenly apart, in how many
distinguishable ways can they be
planted?
Example
The manager of an accounting
department wants to form a threeperson advisory committee from the 16
employees in the department. In how
many ways can the manager do this?
Making It Personal
Write Casa Grande on your paper. How
many distinguishable permutations are
possible?
Try Sacaton.
Write out your entire given name.
How many distinguishable permutations are
possible using the letters in your name?