Combination problems in combinatorics are problems in which the order of the objects doesn't matter. For example, the problem "A class of 65 students can be divided into how many different research teams of seven people each?" is a combination problem; the students can sit in any order within their teams and it won't change the number of possible teams. Using visual representations is not a good system to solve this kind of problem. It is much easier to use a formula.
Permutation problems in combinatorics are problems in which the arrangement of the items does affect the solution. If the question is, "In how many different arrangements can 65 students, divided into research teams of seven students each, choose to sit?", it is a permutation problem; the students within the teams can move around and it won't change the number of possible teams, but it will change their arrangements. A group of seven students can sit in more than one arrangement but only one group. Depending on the scale of the question, permutation problems can be solved with visual representations or with a formula.
A combinatorics problem given as an equation and not as a word problem involves factorials. Factorials are written as a number or variable with an exclamation point after it (5! or x!, for example), and they are equal to that number multiplied by itself and all smaller positive, whole numbers down to 1. 5!, therefore, equals 5 x 4 x 3 x 2 x 1. Solving a combinatorics word problem using a formula also requires the use of factorials.
The standardized test for admission to master's degree programs in business -- the Graduate Management Admission Test, or GMAT -- includes both combination and permutation combinatorics problems. Some other variants are problems with identical objects or circular arrangements. In problems with two or more identical objects, switching these objects doesn't change the arrangements, so there are actually fewer permutations than there seem. Problems with circular arrangements involve literally arranging objects in a circle; this means that turning the entire circle in either direction does not create a new arrangement. Each of these types of problems requires a modification to the basic permutation or combination formulas.