Combinatorics

Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Aspects of combinatorics include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria, finding "largest", "smallest", or "optimal" objects (as in combinatorial designs, extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have (algebraic combinatorics).

The 8 atoms in Acetic acid can be described using the multiset $\{C,C,H,H,H,H,O,O\}$. These letters can be arranged to create 420 "words". Shown is the one that most resembles the commonly used form: $CH_{3}COOH.$

"There is no problem in all mathematics that cannot be solved by direct counting."-Ernst Mach

See also: w:Combinatorial principles and b:Probability/Combinatorics § What_is_combinatorics?

Developed resources

• Combinatorics/Rule of product counts what might be called "decisions". It is still under construction, but quite developed.
  See also: w:Rule of product and b:Probability/The Counting Principle

• Combinatorics/Binomial coefficients introduces binomial coefficients using a tree diagram that recovers Pascal's triangle. A number of advanced topics are then discussed, including Catalan numbers, and Lattice paths

See also: w:Catalan number, w:Lattice path, w:Binomial theorem, w:Multinomial theorem, and b:Probability/Combinatorics § Binomial_and_multinomial_coefficients

Undeveloped resources

The following resources are undeveloped; some are little more than stubs:

1. Combinatorics/About this resource is currently a stub that contains comments that compare the various wikis.
2. Combinatorics/Graph & Ramsey Theory
3. Combinatorics/Structural Algebra (truly a stub)

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