A combination is a selection of all or part of a
set of objects, without regard to the order in which objects are
selected.
For example, suppose we have a set of three letters: A, B, and C.
We might ask how many ways we can select 2 letters from that set. Each possible
selection would be an example of a combination. The complete list of possible
selections would be: AB, AC, and BC.
When statisticians refer to combinations, they use a specific
terminology. They describe combinations as n distinct objects taken r
at a time. Translation: n refers to the number of objects from which the
combination is formed; and r refers to the number of objects used to
form the combination. Consider the example from the previous paragraph. The
combinations were formed from 3 letters (A, B, and C), so n = 3; and
each combination consisted of 2 letters, so r = 2.
Note that AB and BA are considered to be one combination, because
the order in which objects are selected does not matter. This is the key
distinction between a combination and a
permutation.
A combination focuses on the selection of objects without
regard to the order in which they are selected. A permutation, in contrast,
focuses on the arrangement of objects with regard to the order in which
they are arranged.
For an example that counts the number of combinations, see
Sample Problem 2.