MCS 256 Exam 3
Thursday, April 17
-
Exam 3 will be a closed-book examination, but you may use one new page
of handwritten notes (in addition to your previous exam note pages).
-
Exam 3 will consist of problems and questions concerning enumerative
combinatorics corresponding to the material we have covered in class
and in the problem sets.
-
The kinds of problems you should be able to handle ared outlined below.
HOW TO COUNT without counting:
Enumerative Combinatorics
- Basic principles of counting
Notation: The number of elements in a set A is denoted #A or |A|.
- The one-to-one correspondence principle
- The addition principle: counting by cases
- Disjoint sets/cases
- Inclusion-Exclusion Principle
- Tree diagrams
- The multiplication principle: counting by stages
- Exponential formulas: counting functions, subsets
- Counting and probability
- Forms for answers
- Explicit, closed-form formulas
- Recurrences
- 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
distinct elements
- Sampling
The Fourfold Way
- Distributions/functions
- Number of ways to distribute a articles into b
boxes
- Distinguishable articles, distinguishable boxes
- Indistinguishable articles, distinguishable boxes
- Number of ways to distribute a articles into b boxes
with a given number in each box
- Distinguishable articles, distinguishable boxes
Multinomial distribution
Maxwell-Boltzmann statistics
- Indistinguishable articles, distinguishable boxes
Bose-Einstein statistics
Fermi-Dirac statistics
- Number of ways to distribute a articles
into b indistinguishable boxes
- Distinguishable articles: partitions of a set
- Stirling numbers of the second kind
- Bell numbers
- Indistinguishable articles: partitions of an integer
- Generating functions
- Ordinary generating functions