The Multiplication Rule
Lecture 43
Section 9.2
Robb T. Koether
Hampden-Sydney College
Thu, Apr 11, 2013
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The Multiplication Rule
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1
Possibility Trees
2
The Multiplication Rule
3
Permutations
4
r -Permutations
5
Assignment
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The Multiplication Rule
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Outline
1
Possibility Trees
2
The Multiplication Rule
3
Permutations
4
r -Permutations
5
Assignment
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The Multiplication Rule
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Possibility Trees
Definition (Possibility Tree)
A possibility tree is a tree structure that shows all possible sequences
of outcomes in a multi-stage experiment.
Each path (from root to leaf) represents a possible outcome.
The probability of a path is the product of the probabilities that
each branch will be taken along the way.
If the branchings are always equally likely, then the outcomes are
equally likely.
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The Multiplication Rule
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Example
H
H
T
H
H
T
T
H
H
T
T
H
T
T
In the experiment of tossing 3 coins, each coin toss is a separate
stage.
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The Multiplication Rule
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Outline
1
Possibility Trees
2
The Multiplication Rule
3
Permutations
4
r -Permutations
5
Assignment
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The Multiplication Rule
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The Multiplication Rule
Theorem
If an experiment consists of k stages and the i-th stage has ni possible
outcomes, then the experiment has n1 n2 · · · nk possible outcomes.
It is necessary that the number of outcomes at each stage is
independent of the particular outcomes of the earlier stages.
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Example
In how many ways can we select 3 aces from a deck of 52 cards,
assuming that the order matters?
In how many ways can we select 5 red cards from a deck of 52
cards, assuming that the order matters?
In how many ways can we choose three integers, each from 1 to
3, that add up to 6, assuming that the order matters?
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The Multiplication Rule
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Outline
1
Possibility Trees
2
The Multiplication Rule
3
Permutations
4
r -Permutations
5
Assignment
Robb T. Koether (Hampden-Sydney College)
The Multiplication Rule
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Permutations
Definition (Permutation)
A permutation of a set of elements is an arrangement of those
elements in a sequence.
Permutations of 4 objects.
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The Multiplication Rule
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Number of Permutations
Theorem
Given n distinct objects, there are n! permutations of those objects.
Permutations of 4 objects.
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The Multiplication Rule
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Examples
Y
M
R
C
B
G
A traveling salesman (or the UPS man) must visit each of 6 cities.
In how many different ways can he do that?
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Examples
The traveling salesman must begin in city R and finish in city B. In
how many ways can he do that?
The traveling salesman must visit city B immediately after visiting
city R. In how many ways can he do that?
The traveling salesman must visit city B after visiting city R. In
how many ways can he do that?
The traveling salesman must visit cities R, G, and B in that order.
In how many ways can he do that?
The traveling salesman must visit city B after city R and city G
after city Y . In how many ways can he do that?
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The Multiplication Rule
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Outline
1
Possibility Trees
2
The Multiplication Rule
3
Permutations
4
r -Permutations
5
Assignment
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The Multiplication Rule
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r -Permutations
Definition (r -Permutation)
An r -permutation of n elements is an ordered set of r of the n
elements. The number of r -permutations of a set of n elements is
denoted P(n, r ). Define P(n, 0) = 1.
The TI-83 has a function nPr that computes P(n, r ).
For example, the TI-83 computes 6 nPr 2 to be 30.
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r -Permutations
Theorem
For any integer n > 0 and for any integer r with 0 ≤ r ≤ n,
P(n, r ) =
n!
.
(n − r )!
For example,
6!
(6 − 2)!
6!
=
4!
=6·5
P(6, 3) =
= 30.
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The Multiplication Rule
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r -Permutations
Note that P(n, r ) can be defined recursively:
P(n, 0) = 1,
P(n, r ) = nP(n − 1, r − 1) if r ≥ 1.
Prove that P(n, r + 1) = (n − r )P(n, r ).
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The Multiplication Rule
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Outline
1
Possibility Trees
2
The Multiplication Rule
3
Permutations
4
r -Permutations
5
Assignment
Robb T. Koether (Hampden-Sydney College)
The Multiplication Rule
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Assignment
Assignment
Read Sections 9.2, pages 525 - 536.
Exercises 2, 3, 5, 7, 9, 10, 12, 19, 22, 31, 39, 42, page 536.
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The Multiplication Rule
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