UNIT 3--COMBINATORICS & ELEMENTARY PROBABILITY
- SET THEORY
- Set notions and notations. Venn diagrams.
- Set operations: union, intersection, complement, difference
- The correspondence between logic and set theory:
or <--> union, and <--> intersection, not <--> complement, etc.
- HOW TO COUNT
- Basic principles of counting
Notation: The number of elements in a set A is denoted #A or |A|
- The one-to-one correspondence principle
- If A <--> B, then |A| = |B|.
- Tree diagrams: Possibilities <--> paths to leaves <-->
- The addition principle: counting by cases
- Disjoint sets/cases
- Subtraction formula
| Ac | = |U\A| =
|U| - |A| if U = universal set.
- Inclusion-Exclusion Principle
- Tree diagrams: #leaves = sum of #leaves in subtrees.
- The multiplication principle: counting by stages
- Exponential formulas: counting functions, 1-1 functions, subsets
- # functions f: A --> B is |B||A|.
- # 1-1 functions f: A --> B is
- # subsets of set A is 2|A|.
- Division formulas
- # distinguishable or unordered cases from # distinguishable
or ordered cases
- Counting and probability:
P(A) = |A|/|S|.
- Forms for answers
- Explicit, closed-form formulas
- Generating functions
- Key questions
- What would a list of the possibilities look like?
- Is order important?
- What cases are indistinguishable?
- What are the constraints on repetitions?
- What is the answer in small cases?
- Linearly ordered arrangements
Finite sequences, strings, words, ...
- Ordered arrangements without repetitions, permutations
- Ordered arrangements with unlimited repetition allowed
- Ordered arrangements with limited repetition allowed
- Specified numbers of repetitions: permutations of a multiset
- Bounded numbers of repetitions: r-permutations of a multiset
- Other restrictions
- Unordered selections
- Selections without repetitions
Combinations,subsets, samples without replacement
- Selections with unlimited repetition allowed
Multisets, samples with replacement,
nonnegative integer solutions of
x1 + ... + xn = r.
- Selections with limited repetition allowed
Submultisets, r-combinations of a multiset with n
- Number of ways to distribute a articles into b
- Distinguishable articles, distinguishable boxes
- Indistinguishable articles, distinguishable boxes
- Distinguishable articles, indistinguishable boxes:
Partitions of a set
- Stirling numbers of the second kind
- Indistinguishable articles, indistinguishable boxes:
Partitions of an integer
(not on the test)
- Number of ways to distribute a articles into b boxes
with a given number in each box
- Distinguishable articles, distinguishable boxes:
- Indistinguishable articles, distinguishable boxes:
Selections from a multiset
- The probability model
- Sample space
The class of events includes the sample space and
is closed under complements and countable unions and intersections.
- Additivity and countable additivity
- Rules for computing probabilities
- Discrete probability models: the sample space is countable.
- Equally likely (equiprobable) outcomes: P(A) = #A/#S = |A|/|S|.
This relates probability & combinatorics.
- General finite probability models