What is combinatorics?
Combinatorics is a branch of mathematics—overall, called combinatorial mathemat-
ics—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 par-
ticular order, often in a one-to-one correspondence. The number of permutations of a
particularly sized set with nmembers 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 fac-
torials, see “Algebra.”)What is an ordered pair?
An ordered pair is two quantities—usually written as (a, b)—that have a significant
order; thus, (a, b) does not equal (b, a). Ordered pairs are used in set theory to define
members in a function.Ordered pairs are also valuable in linear equations and graphing, in which the x
coordinate is the first number and the ycoordinate is the second number, or (x, y).
They are used on a grid to locate a point. (For more information about ordered pairs
and graphs, see “Geometry and Trigonometry.”)How do functionspertain to sets?
A function in sets pertains to a correspondence between two sets called the domainand
range;each member of the domain has exactly one member of the range. It is often
called a many-to-one (or sometimes one-to-one) relation. For example, f {(1, 2), (3,
6), (4, 2), (8, 0), (9, 6)} is a function, with each set of numbers being an ordered pair.
This is because it assigns each member of the set {1, 3, 4, 8, 9} exactly one value in the
set {2, 6, 2, 0, 6}. It never has two ordered pairs with the same xand different yval-
ues. In this case, the domain is {1, 3, 4, 8, 9} and the range is {2, 6, 2, 0, 6}.
To show an example that is nota function, f {(1, 8), (4, 2), (3, 5), (1, 3), (6, 11)}
is nota function because it does not assign each member of the set exactly one value:
It assigns x1 to both y8 and y3, or it has two ordered pairs that have the
same xvalues to two different yvalues, (1, 8) and (1, 3). (For more information about
128 functions, see “Algebra.”)