Chapter 3 Probability
3-6 Counting
Initial Objective: Develop procedures to determine the #
of elements in a set without resorting to listing them first.
Order is important
Worksheet: Coloring outfits to determine
the total number of combinations possible.
http://illuminations.nctm.org/lessons/combinat
ions/Combinations-AS-ShortsandShirts.pdf
Possible Results

Yellow shirt, Brown shorts
Yellow shirt, Black shorts
Yellow shirt, Green shorts
Yellow shirt, Purple shorts
Orange shirt, Brown shorts
Orange shirt, Black shorts
Orange shirt, Green shorts
Orange shirt, Purple shorts
Blue shirt, Brown shorts
Blue shirt, Black shorts
Blue shirt, Green shorts
Blue shirt, Purple shorts
Red shirt, Brown shorts
Red shirt, Black shorts
Red shirt, Green shorts
Red shirt, Purple shorts
◦ Website Bobby Bear -Students can pick an outfit for
Bobbie Bear and customize the outfit similar to the one
on the activity sheet.

http://illuminations.nctm.org/ActivityDetail.aspx?ID
=3
Discussion Questions

How did your prediction compare to your
actual answer? How do you explain this?

Which method would be more efficient for
finding the total number of outfits:
multiplying, drawing a tree diagram, or
making a table?

Which method would be more useful for
identifying the different combinations
(outfits) possible: multiplying, drawing a tree
diagram, or making a table?
Fundamental / Basic Counting Rule
For a sequence of 2 events in which the 1st event can
occur m ways and the 2nd event can over n ways, the
events together can occur a total of m x n ways.
Example
In designing a computer, if a byte is defined to be a
sequence of 8 bits, and each bit must be a 0 or 1, how
many different bytes are possible?
Answer
Example
 In designing a computer, if a byte is
defined to be a sequence of 8 bits, and
each bit must be a 0 or 1, how many
different bytes are possible?

Answer: Since each bit can occur in 2
ways (0 or 1) and we have a sequence of
8 bits, the total # of different possibilities
is given by 2x2x2x2x2x2x2x2= 256
Factorial Rule- n different items can be
arranged in order n! different ways
Ex. 5! = 5x4x3x2x1= 120
0! =1
 Key is on your calculator

Note:The factorial rule reflects that fact that the
1st item maybe selected n different ways; the 2nd
item maybe selected n-1 ways, and so on.
Factorial Examples

Example: How many possible ways
routes are there to 3 different cities?

Example: What about possible routes
each of the 50 states?
What if you don’t what to
include all of the items available?
Order is still important!
Permutations Rule
The # of permutations(or arrangements) of
r items selected from n available items is
n pr =n!/(n-r)!
Note: rearrangements of the same items to be different.
Look at page 155 for an example.
What if we tend to select r items
from n available items, but do not
take order into account?
◦ We are really concerned with possible
combinations.
B: Order is not important
When different ordering of the same items
are to be counted separately, we have a
permutation problem,
 but when different orderings are not to be
counted separately, we have a combination
problem.

Combinations Rule
The # of combinations of r items
selected from n available items is
nCr =
[Other notations for nCr are
Run through page 156 to 158
Could you image not having these
counting techniques,
it would take hours and hours to come up
with all the possibilities.