Combinatorics is the study of counting and arranging objects.
Combinatorial problems often involve selecting objects from a set, arranging objects in a sequence, or constructing objects subject to certain conditions.
Permutations
A permutation is an arrangement of objects in a specific order.
The number of permutations of n objects is n!, i.e., the product of all positive integers up to n.
Combinations
A combination is a selection of objects from a set without regard to order.
The number of combinations of k objects from a set of n objects is denoted by n choose k, written as nCk or C(n,k), and is given by the formula n! / (k!(n-k)!).
Pigeonhole Principle
The pigeonhole principle states that if k + 1 or more objects are placed into k boxes, then there must be at least one box containing two or more objects.
Inclusion-Exclusion Principle
The inclusion-exclusion principle is a counting technique used to calculate the size of a union of sets.
For two sets A and B, the principle states that
A union B
=
A
+
B
-
A intersect B
.
The principle can be extended to more than two sets.
Generating Functions
A generating function is a formal power series used to encode information about a sequence of numbers.
The coefficients of the generating function can be used to determine the number of ways to count or arrange objects.
Generating functions can be manipulated using algebraic operations such as addition, multiplication, and composition.
Recurrence Relations
A recurrence relation is an equation that recursively defines a sequence of numbers.
Recurrence relations can be used to model combinatorial problems, such as counting the number of ways to arrange objects subject to certain conditions.
Techniques such as substitution and generating functions can be used to solve recurrence relations.