Thinking
Mathematically
Counting Methods and Probability
11.1 The Fundamental Counting Principle
The Fundamental Counting Principle
If you can choose one item from a group of M
items and a second item from a group of N items,
then the total number of two-item choices is M 
N.
Exercise 11.1 #3
A popular brand of pen is available in three colors (red, green, or
blue) and four writing tips (bold, medium, fine, or micro). How
many different choices of pens do you have with this brand?
The Fundamental Counting
Principle
The number of ways a series of successive things can
occur is found by multiplying the number of ways in
which each thing can occur.
Example: The Fundamental
Counting Principle
Exercise 11.1 #9
A restaurant offers the following lunch menu
Main Course Vegetables
Beverages
Desserts
Ham
Potatoes
Coffee
Cake
Chicken
Peas
Tea
Pie
Fish
Green beans
Milk
Ice cream
Beef
Soda
If one item is selected from each of the four groups, in how
many different ways can a meal be ordered?
Example Telephone Numbers in the
United States
Exercise Set 11.1 #17
In the original (1945) plan for area codes, the first
digit could be 2 through 9, the second 0 or 1, and
the third any digit any number except 0. How
many different area codes are possible under this
plan?
Thinking
Mathematically
Counting Methods and Probability
11.2 Permutations
Permutations
• No item is used more than once.
• The order of arrangement makes a
difference.
Examples: Permutations
Exercise 11.2 #7, 11
You need to arrange nine of your favorite books
along a shelf. How many different ways can you
arrange the books, assuming that the order of the
books makes a difference?
How many ways can five movies be ordered into two
hour time slots between 6PM and 4AM. Two of
the movies have G ratings and are to be shown in
the first two slots. One is rated NC-17 and is to be
shown in the last time slot.
Factorial Notation
If n is a positive integer, the notation n! is the product
of all positive integers from n down through 1.
n! = n(n-1)(n-2)…(3)(2)(1)
0!, by definition is 1.
0!=1
Exercise 11.2 #15
29!
25!
Permutations of n Things Taken r at a
Time
The number of permutations possible if r
items are taken from n items:
Exercise 11.2 #35, 45
Evaluate 8P5
In a race in which six automobiles are entered and there are no ties, in how
many ways can the first three finishers come in ?
Thinking
Mathematically
Counting Methods and Probability
11.3 Combinations
A combination of items occurs when:
• The items are selected from the same
group.
• No item is used more than once.
• The order of the items makes no
difference.
• How is this different from a
permutation?
Permutations or Combinations?
Exercise Set 11.3 #1, 3
• A medical researcher needs 6 people to test the
effectiveness of an experimental drug. If 13 people
have volunteered for the test, in how many ways can 6
people be selected?
• How many different four-letter passwords can be
formed from the letters A, B, C, D, E, F, and G if no
repetition of letters is allowed?
Combinations of n Things Taken r at a
Time
The number of possible combinations if r items are
taken from n items is
n!
n Cr 
(n  r )!r!
Exercise 11.3 #9
Evaluate 11C4
Thinking
Mathematically
Chapter 11
Counting Methods and Probability