cs2102: Discrete Mathematics
Class 25: Counting
David Evans, Mohammad Mahmoody
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Plan
Today: How to Count! (parts of Chapter 15)
Thursday: Probability (parts of Chapters 17,18)
Next Tuesday: Review
Next Thursday: Final Exam
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Examples
1. How many subsets does have if = 20?
2. How many numbers of (at most) 20 bits (in base 2) ?
3. How many (at most) 16-bit numbers with exactly 4 ones?
4. How many ways to choose 12 doughnuts from 5 varieties.
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Why Counting?
• Estimating time/storage/cost of an algorithm
• Estimating the “security”:
how many bits does adversary have to “guess” ?
• Basis for probability theory.
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Recall Finite Cardinality (class 9)
• Many counting problems can be cast as:
“how many elements are in set ?”
The cardinality of the set
= ∈ ℕ ∧ < }
is . If there is a bijection between and
then has the same cardinality.
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Examples
1. How many subsets does have if = 20?
2. How many numbers of (at most) 20 bits?
• Bijection between them:
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Example: number of sequences
• How many -sequences with elements in {1, … , } ?
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• What if 1
and 2
can have intersection?
• What if = 3?
• Generalization: Inclusion-Exclusion (look it up in the book!)
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Example: counting passwords
• Suppose valid password:
1. Sequence of 6 to 8 symbols.
2. First symbol: a letter (lowercase or uppercase).
3. Remaining symbols: must be either letters or digits.
• How many different passwords are possible?
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• is a set containing length- sequences. If
– 1
possibilities for 1st entry
– For each 1st entry: 2
possibilities for 2nd entry
– …
– For each ( − 1)st entry:
possibilities for th entry
• Then = 1
⋅ 2
… ⋅
Generalized Product Rule
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Example: counting permutations
• A permutation of = {1
, …
} is an (ordered) sequence
1
, … ,
such that every
appears exactly once.
• How many permutations are there?
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More on “!”
• 0! = 1
• Approximating !
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• How many ways to select -sequences with
elements chosen from 1, … , :
– Repetition allowed:
– Repetition not allowed?
Role of Repetition (allowed or not?)
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Counting when order does not matter
• How many ways to select a -subset of 1, … , ?
• Namely:
= { ∣ ⊆ , = } what is
?
• This number is denoted as
: read “n choose k”.
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Counting when order does not matter
• How many ways to select a -subset of 1, … , ?
• Namely:
= { ∣ ⊆ , = } what is
?
• Idea: define
to contain -sequences using 1, … , …