Class 25

Counting 1

Transcript

1. cs2102: Discrete Mathematics Class 25: Counting David Evans, Mohammad Mahmoody
2. Plan Today: How to Count! (parts of Chapter 15) Thursday: Probability (parts of Chapters 17,18) Next Tuesday: Review Next Thursday: Final Exam
3. 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.
4. Why Counting? • Estimating time/storage/cost of an algorithm • Estimating the “security”: how many bits does adversary have to “guess” ? • Basis for probability theory.
5. 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.
6. Examples 1. How many subsets does have if = 20? 2. How many numbers of (at most) 20 bits? • Bijection between them:
8. Example: number of sequences • How many -sequences with elements in {1, … , } ?
9. • What if 1 and 2 can have intersection? • What if = 3? • Generalization: Inclusion-Exclusion (look it up in the book!)
10. 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?
11. • 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
12. Example: counting permutations • A permutation of = {1 , … } is an (ordered) sequence 1 , … , such that every appears exactly once. • How many permutations are there?
13. More on “!” • 0! = 1 • Approximating !
14. • How many ways to select -sequences with elements chosen from 1, … , : – Repetition allowed: – Repetition not allowed? Role of Repetition (allowed or not?)
15. 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”.
17. 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, … , …