Lesson 10.5


The numerator and denominator of a
theoretical probability are numbers of
possibilities.
Sometimes those possibilities follow regular
patterns that allow you to “count” them.


The numerator and denominator of a
theoretical probability are numbers of
possibilities.
Sometimes those possibilities follow regular
patterns that allow you to “count” them.


Suppose you want to create a random playlist
from a library of songs on an MP3 player. If
you do not repeat any songs, in how many
different orders do you think the songs could
be played?
In this investigation you will discover a
pattern allowing you to determine the
number of possible orders without listing all
of them.

Step 1 Start by investigating some simple cases.
Consider libraries of up to five songs, and
playlists of up to five of those songs.
◦ Notice n represents the number of songs in the library
(1≤n≤5) and r represents the length of the playlist
(1≤r≤n).

Step 1 Start by investigating some simple cases.
Consider libraries of up to five songs, and
playlists of up to five of those songs.
◦ Suppose we wanted to make playlists of two songs
from a list of three songs. Let A, B, and C represent the
three songs available.
◦ Write out the all the playlists that could be formed.
◦ AB, AC, BA, BC, CA, and CB.
◦ How do you know you have all the ways listed?
Start with n = 1, 2, and 3 and write out all the
playlists you can form
1
2
2
3
6
6
Describe any patterns you found in either the rows
or columns of the table.
Have you used a tree
diagram?
You have
• four choices of sweaters
• six different pants
• two pairs of shoes
How many different outfits are
there?
Why do we multiply?
We can think about using the Counting Principal
We are going to put outfits together with a
sweater, a pair of pants and a pair of shoes. A
box could be used for each item
How many choices do I have for each box?
Why should I multiply these numbers?
4
X
4
X 2
=32
Finish the chart for the number of playlists
1
2
2
3
6
6
4
5
12
20
24
24
60
120
120
Describe any patterns you found in either the rows
or columns of the table.
 Write
an expression for the
number of ways to arrange 10
songs in a playlist from a
library of 150 songs.
150x149x148x147x146x145x144x143x142x141
Suppose there are n1 ways to make a choice, and
for each of these there are n2 ways to make a
second choice, and for each of these there are n3
ways to make a third choice, etc. The product of
n1 ● n2● n3● …..
gives the number of possible outcomes.

Suppose a set of license plates has
any three letters from the alphabet,
followed by any three digits.
◦ How many different license plates are
possible?
◦ What is the probability that a license plate
has no repeated letters or numbers?
___ ___
26
26 ___
26
___
10 ___
10___
10
17, 576, 000 license plates
◦ What is the probability that a
license plate has no repeated letters
or numbers?
26 __
25 __
24 __
10 __9 __8
__
=11,232,000
About 63.9% of license plates would not
have repeated letters or numbers.
When the objects cannot be used more than
once, the number of possibilities decreases at
each step.
These are called “arrangements without
replacement.” This arrangement is called a
permutation.
nPr
is the number of permutations of n things
chosen r at a time.

Seven flute players are performing in an
ensemble.
◦ The names of all seven players are listed in
the program in random order. What is the
probability that the names are in alphabetical
order?
◦ After the performance, the players are
backstage. There is a bench with room for
only four to sit. How many possible seating
arrangements are there?
◦ What is the probability that the group of four
players is sitting in alphabetical order?

Seven flute players are performing in an ensemble.
◦ The names of all seven players are listed in the program in
random order. What is the probability that the names are in
alphabetical order?
7P7
=7 •6 • 5 • 4 • 3 • 2 • 1= 5040
Only one of these arrangements is in
alphabetical order. The probability is 1/5040
or approximately 0.0002.

Seven flute players are performing in an ensemble.
◦ After the performance, the players are backstage. There is a
bench with room for only four to sit. How many possible
seating arrangements are there?
7P4
=7•6•5•4=840

Seven flute players are performing in
an ensemble.
◦ What is the probability that the group of
four players is sitting in alphabetical order?
With each arrangement of 4, there is only one
correct order.
4P4
=4• 3 • 2 • 1 = 24
1/24, or approximately 0.04167.
Notice that the answer to part c does not depend on
the answer to part b
n Pr 
n!
(n  r )!