Transcript Chapter 8 Counting Principles: Further Probability Topics

Chapter 8
Counting Principles:
Further Probability Topics
Section 8.1
The Multiplication Principle;
Permutations
Warm – Up for Sections 8.1 and 8.2
A certain game at an amusement card consists of a
person spinning a spinner, choosing a card, and
then tossing an unbiased coin. Prizes are awarded
based on the combination created from
performing each of the three tasks.
The spinner has three equal areas represented by
Purple, Gold, and Red; the cards to choose from
include a King, Queen, and Joker; and the coin has
a Crown on one side and a Donkey on the other.
How many possible outcomes are there?
If the order in which the tasks were performed
made a difference, would there be more outcomes
or fewer outcomes?

Alice can’t decide what to wear between a pair
of shorts, a pair of pants, and a skirt. She has
four tops that will go with all three pieces: one
red, one black, one white, and one striped.
How many different outfits could Alice create
from these items of clothing?
Bottom
Shorts
Pants
Skirt
Top
Red
Black
White
Striped
Outfit
Shorts, Red Top
Shorts, Black Top
Shorts, White Top
Shorts, Striped Top
Red
Black
White
Striped
Red
Black
White
Striped
12 outfits!!
If the tree diagram is finished, how many outfits will she have?

Tree diagrams are not often convenient, or
practical, to use when determining the number
of outcomes that are possible.

Rather than using a tree diagram to find the
number of outfits that Alice had to choose
from, we could have used a general principle of
counting: the multiplication principle.
Multiplication Principle

Alice can’t decide what to wear between a pair
of shorts, a pair of pants, and a skirt. She has
four tops that will go with all three pieces: one
red, one black, one white, and one striped.
How many different outfits could Alice create
from these items of clothing?
Using the multiplication principle, we multiply the number
of options she has for what to wear on “bottom” and the
number of options she has for what to wear on “top”.
3 bottoms • 4 tops = 12 outfits

A product can be shipped by four airlines and
each airline can ship via three different routes.
How many distinct ways exist to ship the
product?

How many different license plates can be made if
each license plate is to consist of three letters
followed by three digits and replacement is allowed?
26 • ___
26 • ___
26 • ___
10 • ___
10 • ___
10 = 26 ³ • 10 ³
___
L
L
L
D D
D
= 17, 576, 000

If replacement is not allowed?
26
25
24
10
9
8
___ • ___ • ___ • ___ • ___ • ___
L
L
L
D D D
= 11, 232, 000
How many different license plates can be made
if each license plate begins with 63 followed by
three letters and two digits?
How many different social security numbers are
possible if the first digit may not be zero?
Marie is planning her schedule for next semester. She must
take the following five courses: English, history, geology,
psychology, and mathematics.
a.) In how many different ways can Marie arrange
her schedule of courses?
b.) How many of these schedules have mathematics
listed first?
You are given the set of digits {1, 3, 4, 5, 6}.
a.) How many three-digit numbers can be formed?
b.) How many three-digits numbers can be formed if
the number must be even?
c.) How many three-digits numbers can be formed if
the number must be even and no repetition of
digits is allowed?
A certain Math 110 teacher has individual photos of
each of her three dogs: Indy, Sam, and Jake. In
how many ways can she arrange these photos in a
row on her desk?
Factorial Notation
If seven people board an airplane and there are
nine aisle seats, in how many ways can the
people be seated if they all choose aisle seats?
Permutations




A permutation of r (where r ≥ 1) elements from
a set of n elements is any specific ordering or
arrangement, without repetition, of the r
elements.
Each rearrangement of the r elements is a
different permutation.
Permutations are denoted by nPr or P(n, r)
Clue words: arrangement, schedule, order, awards,
officers

A disc jockey can play eight records in a 30minute segment of her show. For a particular
30-minute segment, she has 12 records to select
from. In how many ways can she arrange her
program for the particular segment?

A chairperson and vice-chairperson are to be
selected from a group of nine eligible people.
In how many ways can this be done?
Distinguishable Permutations

If the n objects in a permutation are not all
distinguishable – that is, if there so many of
type 1, so many of type 2, and so on for r
different types, then the number of
distinguishable permutations is
n!
.
1
2
r
n ! n ! ••• n !
How many distinct arrangements can be formed
from all the letters of SHELTONSTATE?
Step 1: Count the number of letters in the word, including repeats.
12 letters
Step 2: Count the number of repetitious letters and the number of
times each letter repeats.
S : 2 repeats
Solution:
E : 2 repeats
12!
2! 2! 3!
.
T : 3 repeats
= 19, 958, 000

In how many distinct ways can the letters of
MATHEMATICS be arranged?

In how many distinct ways can the letters of
BUCCANEERS be arranged?