To keep track of which and how many combinations exist, it can be helpful to use a counting model.

Counting Tree

A counting tree is a model that provides an overview of the number of combinations.

The counting tree works by adding a branch each time a combination is made. Once the tree is complete, you count the number of endpoints on the tree to find the result of the problem.

Example: What possibilities do you have when flipping two coins (heads/tails)?

Matrix

Another way to model combinations is using a matrix. A matrix is a two-dimensional table. In some cases, a matrix is advantageous because possible combinations are easier to read since they are listed directly in the matrix. However, the matrix has the limitation that it can only work in two dimensions. For example, you can create a matrix for tossing two dice, but not for three or more.

Example: A matrix for two coin flips (with replacement):

From both the counting tree and matrix, you can see that there are 4 possible outcomes (Heads-Heads, Heads-Tails, Tails-Heads, and Tails-Tails).

For example, you can see that there are 3 possibilities out of 4 where at least one heads is present.