Foundations of Mathematics

Set Theory

What is combinatorics?

Combinatorics is a branch of mathematics—overall called combinatorial mathematics—that studies the enumeration, combination, and permutation of sets and the mathematical relations that involve these properties, defined as:

Enumeration—Sets can be identified by the enumeration of their elements; in other words, determining (or counting) the set of all solutions to a given problem.

Combination—Combination is how to count the many different ways elements from a given set can be combined. For example, the 2-combinations of the 4-set {A,B,C,D} are {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}.

Permutation—Permutation is the rearrangement of elements of a set into a particular order, often a one-to-one correspondence. The number of permutations of a particularly sized set with n members is written as the factorial n!. For example, a set with 4 members would have 4 in first place to 1 in the last place. This would equal 4 × 3 × 2 × 1 = 4!, or 24, permutations of 4 members. (For more information about factorials, see “Algebra.”)


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