Counting Principles

 In
this section, you will study several
techniques for counting the number of
ways an event can occur. One is the
Fundamental Counting Principle. You can
use this principle to find the number of
ways two or more events can occur in
If one event can occur in m ways and a second
event can occur in n ways, the number of ways
the two events can occur in sequence is m*n. This
rule can be extended for any number of events
occurring in sequence.
 Using the Principle - You are purchasing a new
car. Using the following manufacturers, car sizes,
and colors, how many different ways can you
select one manufacturer, one car size, and one
• Manufacturer: Ford, GM, Chrysler
• Car size: small, medium
• Color: white (W), red (R), black (B), green (G)
There are three choices of manufacturers, two car sizes, and four colors. So, the
number of ways to select one manufacturer, one car size, and one • color is
3 · 2 ·4 = 24 ways.
A tree diagram can help you see why there are 24 options.
 Your
choices now include a Toyota, a
large car, or a tan or gray car. How many
different ways can you select one
manufacturer, one car size, and one
 The
access code for a car's security
system consists of four digits. Each digit
can be 0 through 9. How many access
codes are possible if
• 1. each digit can be used only once and not
• 2. each digit can be repeated?
 How
many license plates can you make if
a license plate consists of
• 1. six (out of 26) alphabetical letters each of
which can be repeated?
• 2. six (out of 26) alphabetical letters each of
which cannot be repeated ?
An important application of the Fundamental
Counting Principle is determining the number of
ways that n objects can be arranged in order or
in a permutation .
 A permutation is an ordered arrangement of
objects. The number of different permutations of
n distinct objects is n! .
 The expression n! is read as n factorial and is
defined as follows.
n! = n*(n -1) '(n - 2) ' (n - 3) .. . 3 * 2’ * 1
 As a special case,
O! = 1.
 Here
are several other values of n!
• 1! = 1
• 2! = 2·1 = 2
• 3!=3·2·1=6
• 4! = 4·3·2·1 = 24
• 5! = 5 . 4 . 3 . 2 . 1 = 120
 The
starting lineup for a baseball team
consists of nine players. How many
different batting orders are possible
using the starting lineup?
 The
number of permutations is
9! = 9·8·7·6·5·4·3·2·1 = 362,880.
So, there are 362,880 different batting
 Suppose
you want to choose some of the
objects in a group and put them in order.
Such an ordering is called a permutation
of n objects taken, r at a time.
 The number of permutations of n distinct
objects taken r at a time is
nPr=-n! /( n-r)!, where r ≤ n.
 Find
the number of ways of forming threedigit codes, which no digit is repeated.
 Solution
– to form a three digit code in which
no digit is repeated, you need to select 3
digits from a group of 10, so n=10 and r=3
So there are 720 possible three digit codes that do not have repeating digits.
 In
a race with eight horses, how many
ways can three of the horses finish in first,
second, and third place? Assume that
there are no ties.
• Find the quotient of n! and (n - r)! . (List the
factors and divide out.)
• Write the result as a sentence.
Forty-three race cars started the 2004 Daytona 500. How many
ways can the cars finish first, second, and third?
You need to select three race cars from a group of 43, so n = 43 and
r = 3. Because the order is important, the number of ways the cars
can finish first, second, and third is
The board of directors for a company has 12 members.
One member is the president, another is the vice
president, another is the secretary, and another is the
treasurer. How many ways can these positions be
• Identify the total number of objects n and the
number of objects r being chosen in order.
• Evaluate nPr
You want to buy three CDs from a selection of
five CDs. There are 10 ways to make your
In each selection, order does not matter (ABC is
the same set as BAC). The number of ways to
choose, r objects from n objects without regard to
order is called the number of combinations of n
objects taken r at a time.
combination is a selection of ,. objects
from a group of n objects without regard
to order and is denoted by nCr, The
number of combinations of r objects
selected from a group of n objects is
A state's department of transportation plans to
develop a new section of interstate highways and
receives 16 bids for the project. The state plans
to hire 4 of the bidding companies. How many
different combinations of four companies can be
selected from the 16 bidding companies?
 The state is selecting four companies from a
group of 16, so n=16 and r =4. Because order is
not important, there are
different combinations.
 The
manager of an accounting department
wants to form a three-person advisory
committee from the16 employees in the
department. In how many ways can the
manager form this committee?
• Identify the number of objects in the group n and
the number of objects to be selected r.

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