Download DISCRETE MATH: LECTURE 20 1. Chapter 9.2 Possibility Trees

DISCRETE MATH: LECTURE 20
DR. DANIEL FREEMAN
1. Chapter 9.2 Possibility Trees and the Multiplication Rule continued...
Theorem 1.1 (Equally Likely Probability Formula). If S is a finite sample space with
N (S) elements in which all outcomes are equally likely and E ⊆ S is an event with N (E)
elements, then the probability of E is:
N (E)
P (E) =
N (S)
Theorem 1.2 (The Multiplication rule).
If an operation consists of k steps and
the 1st step can be performed in n1 ways,
the 2nd step can be performed in n2 ways,
...
the kth step can be performed in nk ways,
then the entire operation can be performed in n1 n2 ...nk ways.
Example: Number of elements in a Cartesian product. Let A1 , A2 , ..., Ak be sets. Assume
A1 contains n1 elements, A2 contains n1 elements,... Ak contains nk elements.
A1 × A2 × ... × Ak = {(a1 , a2 , ..., ak ) | a1 ∈ A1 , a2 ∈ A2 , ..., ak ∈ Ak }
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DR. DANIEL FREEMAN
1.1. Permutations.
Definition 1.3. Let A be a finite set. A permutation of A is a linear ordering of the
elements of A.
Example: A = {a, b, c}
Theorem 1.4. For any integer n ≥ 1, the number of permutations of a set with n elements
is n!.
Example: Permutations of objects around a circle. How many ways are there to arrange the letters {a, b, c, d, e} into a circle if only relative position matters? That is an
arrangement into a circle is equivalent to any rotation of itself.
DISCRETE MATH: LECTURE 20
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Definition 1.5. Let r and n be positive integers with r ≤ n. An r-permutation of a
set of n elements is a linear ordering of a subset consisting of r elements. The number of
r-permutations of a set of n elements is denoted P (n, r).
Example: A = {a, b, c}. The 2-permutations of A are:
Theorem 1.6. For any integers n ≥ r, the number of r-permutations of a set with n
elements is:
n!
P (n, r) = n(n − 1)(n − 2)...(n − r + 1) =
(n − r)!
Example: How many 3 letter words can be made from the letters {C, O, W, S}? (they
don’t have to be real words, just 3 letters in a row)
How many 3 letter words can be made from the letters {C, O, W, S}, if the first letter of
the word is C?
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DR. DANIEL FREEMAN
In class problems:
(1) List the functions from the set {a, b} to the set {1, 2, 3}. How many are there?
(2) If A is a set with n elements and B is a set with m elements, how many functions
f : A → B exist? Use the multiplication rule to explain why.
(3) List the one to one functions from the set {a, b} to the set {1, 2, 3}.
(4) If A is a set with n elements and B is a set with m elements, how many one to one
functions f : A → B exist? Use the multiplication rule to explain why. (a function
f is one to one if x = y ⇔ f (x) = f (y))
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(5) List the subsets of {a, b} and {a, b, c}. How many are there?
(6) How many subsets does a set of n elements have? Use the multiplication rule to
explain why.
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DR. DANIEL FREEMAN
2. 9.3 Counting Elements of Disjoint Sets: The Addition Rule
If you separate a collection of objects into disjoint piles then the total number of objects
is the sum of the number of objects in each pile.
Theorem 2.1 (The Addition Rule). Suppose A is a finite set and equals the union of k
disjoint subsets A1 , A2 , ..., Ak . Then
N (A) = N (A1 ) + N (A2 ) + ... + N (Ak )
Or in other words, if A1 , A2 , ..., Ak are pairwise disjoint then
N (∪ki=1 Ai )
=
k
X
N (Ai ).
i=1
Example: A password will consist of an ordered list of at most 4 numbers from 0 to 9.
List some examples. How many possible passwords exist?