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Combinatorics

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Combinatorics is an area of discrete mathematics that studies collections of distinct objects and the ways that they can be counted or ordered, or used to satisfy some optimality criterion.

The most basic ideas in combinatorics include:

factorials
The number of possible arrangements of n distinct items is n-factorial, written n! , which equals
n\times(n-1)\times(n-2)\times\cdots\times2\times1
  • Example: Three items, A, B, and C, can be arranged in 3!=3\times2\times1=6 different ways: ABC, ACB, BAC, BCA, CAB, and CBA.
permutations
The number of arrangements that are possible when a subset of k items is taken from a set of n distinct items is a "permutation of n objects taken k at a time", which can be written as P^n_k or {}_nP_k , and is equal to \frac{n!}{(n-k)!} .
  • Example: The number of possible arrangements of the four letters A, B, C, D, taken two at a time, is P^4_3=\frac{4!}{(4-2)!}=\frac{24}{2}=12: AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, and DC.
combinations
The number of possible subsets of k items taken from a set of n items, where the order of the items doesn't matter (e.g., the sets ABC and BCA are considered equivalent), is a "combination of n objects taken k at a time", which is written C^n_k or {}_nC_k or \binom{n}{k} , and is equal to \frac{n!}{k!(n-k)!} .
  • Example: The number of subsets of two letters chosen from the four letters A, B, C, and D, is C^4_2=\frac{4!}{2!(4-2)!}=\frac{24}{2\times2}=6 : AB, AC, AD, BC, BD, and CD.
distributions
partitions of integers or of sets
recurrence relations
inclusions
inversions
inclusion/exclusion principle
derangements and subfactorials
repetitions and replacements
various restrictions placed on problems
fundamental counting principle
circular permutations
generating functions
free and fixed permutations, rotational symmetry and reflective symmetry
cyclic permutations
multisets
Pascal's Triangle

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