Counting Methods and
Permutations
Example
Suppose
that Maggie has 5 pairs of jeans
and 6 T-Shirts. How many outfits
consisting of one pair of jeans and one Tshirt does she have available to her
Counting Principle
If
there are m ways to make a selection and
n ways to make a second selection, then
there are m x n ways to make the pair of
selections.
Counting Methods and
Permutations
Suppose you have three shirts and three ties that coordinate. Make a
tree diagram to find the number of possible outfits you have.
Ties
Outfits
Shirts
Shirt 1
Tie1
Tie 2
Tie 3
Shirt 1, Tie 1
Shirt 1, Tie 2
Shirt 1, Tie 3
Shirt 2
Tie1
Tie 2
Tie 3
Shirt 2, Tie 1
Shirt 2, Tie 2
Shirt 2, Tie 3
Tie1
Shirt 3, Tie 1
Shirt 3
Tie 2
Shirt 3, Tie 2
Tie 3
Shirt 3, Tie 3
There are nine possible outfits.
Example
In
how many different ways can the letters
MATH be arranged?
Permutation
Of
the members in a set of objects is any
arrangement of those objects in a specific
order.
Example
How
many different arrangements of the
letters in EIGHT are there if all the letters
are used exactly once?
5!
= 5x4x3x2x1 (this is called factorial
notation)
Another notation would be 5P5
Simplify 8P5.
Counting Methods and
Permutations
Method 1: Use pencil and paper.
8P5 =
8•7•6•5•4
= 6720
The first factor is 8 and there are 5 factors.
Simplify.
Method 2: Use a graphing calculator.
Enter the first factor,8.
Use MATH to select nPr in the PRB screen.
Input 5, since there are 5 factors. Press enter.
8P5 =
6720
Counting Methods and
Permutations
Suppose you use five different letters from the 26 letters of
the alphabet to make a password. Find the number of possible fiveletter passwords if letters cannot repeat.
There are 26 letters in the alphabet. You are finding the number of
permutations of 26 letters arranged 5 at a time.
26P5 =
26 • 25 • 24 • 23 • 22
Use a calculator.
= 7,893,600
There are 7,893,600 five-letter passwords in which letters do not repeat.
Permutations with Repetition
How
many ways can you arrange the word
SEE?
Another
way would be 3!
2! (two E’s repeat)
Homework
p702-703
#2,6,7-16all, 26