Fundamental Counting Principle, p' 558578 11'111'3

1
Fundamental Counting Principle, p.
558-578 (11.1-11.3)

• OBJECTIVES
• Use Fundamental Counting Principle to determine
  the number of possible outcomes
• Use Principle to count permutations
• Distinguish between permutation and combination
  problems

2
The Fundamental Counting Principle, p. 560

• 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.

3

• This is the semester that you decide to take your
  required psychology and social science courses.
• 15 sections of psychology from which you can
  choose.
• 9 sections of social science that are available
  at times that do not conflict with those for
  psychology.
• In how many ways can you create two-course
  schedules that satisfy the psychology-social
  science requirement?

For both courses you have 15 ? 9, or 135
choices.
4
The Fundamental Counting Principle, p. 561

• 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.

Next semester you are planning to take three
courses - math, English, and humanities. 8
sections of math 5 of English 4 of humanities
that you find suitable. Assuming no scheduling
conflicts, there are 8 ? 5 ? 4 160 different
three course schedules.
5

• A multiple-choice test that has ten questions.
• Each question has four choices, with one correct
  choice per question.
• If you select one of these options per question
  and leave nothing blank, in how many ways can you
  answer the questions?

Multiply the number of choices, 4, for each of
the ten questions 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4 ? 4 ?
4 ? 4 1,048,576
6

• United States telephone numbers have three-digit
  area codes followed by seven-digit local
  telephone numbers.
• Area codes and local telephone numbers cannot
  begin with 0 or 1.
• How many different telephone numbers are possible?

8 ? 10 ? 10 ? 8 ? 10 ? 10 ? 10 ? 10 ? 10 ? 10
6,400,000,000
7
Permutations, p. 565

• No item is used more than once.
• The order of arrangement makes a difference.

Arrange seven books along a small shelf. How many
different ways can you arrange the books,
assuming that the order of the books makes a
difference to you? 7 ? 6 ? 5 ? 4 ? 3 ? 2 ? 1
5040 There are 5040 different possible
permutations.
8
Factorial Notation, p. 567

• 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

9
Permutations of n Things Taken r at a Time, p. 569

• The number of permutations possible if r items
  are taken from n items

10
A combination (p. 573) of items occurs when

• The item are selected from the same group.
• No item is used more than once.
• The order of the items makes no difference.

11
Example Distinguishing between Permutations and
Combinations

• Six students are running for student government
  president, vice-president, and treasurer.
• The student with the
• greatest number of votes becomes the president,
• the second biggest vote-getter becomes
  vice-pres.,
• and the student who gets the third largest
  number of votes becomes student government
  treasurer.
• How many different outcomes are possible for
  these three positions?

The order matters. This is a problem involving
permutations.
12
Example Distinguishing between Permutations and
Combinations

• Six people are on the volunteer board of
  supervisors for your neighborhood park.
• A three-person committee is needed.
• How many different committees could be formed
  from the six people on the board of supervisors?

13
Solution

• The order in which the three people are selected
  does not matter because they are not filling
  different roles on the committee.
• Because order makes no difference, this is a
  problem involving combinations.

14
Example Distinguishing between Permutations and
Combinations

• Baskin-Robbins offers 31 different flavors of ice
  cream.
• A bowl consists of three scoops of ice cream,
  each a different flavor.
• How many such bowls are possible?

15
Solution

• The order in which the three scoops of ice cream
  are put into the bowl is irrelevant.
• Different orderings do not change things, and so
  this problem is combinations.

16
Combinations of n Things Taken r at a Time, p. 575
17
HOMEWORK

• Office Hours MWF 9-945
• T/Th 130-330
• Tutoring M-Th 4-6
• p.564 1-21alternate odd
• p. 571-572 1-56 alternate odd
• p. 578-579 1-40 alternate odd