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Organized Counting

Organized counting is a method used in combinatorics to systematically count the number of ways an event can occur. By structuring counting methods logically, we can avoid errors and ensure we consider all possible outcomes. The techniques and mathematical logic for counting possible arrangements or outcomes are useful for a wide variety of applications. Some examples would be a conference manager arranging a schedule of seminars, or a school board trying to make the most efficient use of its buses.

Combinatorics and Tree Diagrams

Combinatorics is the branch of mathematics dealing with counting, arranging, and selecting objects. These techniques are useful for probability calculations. One useful tool in combinatorics is a tree diagram, which visually represents all possible outcomes of a situation by branching from one stage to the next.

A Tree Diagram representing the different combinations of main courses and sides.
Fig.1 - Tree diagram for ordering a meal


As you can see here, we have two Main dishes, Chicken or Beef, and each of the two main dishes have two side dishes, Rice or Fries. By using the multiplicative counting principle, we have 4 different possibilities of a meal combination. But, how did we get that number in the first place, and what is the Multiplicative Counting Principle?


In the above tree, how many possibilites of meal combinations are there?

By looking at the tree diagram above, we can identify there are 4 possible meal options. There is a systematic way to figure it out, which we will cover in the next section.

Multiplicative Counting Principle

The multiplicative counting principle states If a task or process is made up of stages with separate choices:

\(\text{Outcomes} = m * n * p ...\)

Where \(m\) is the number of choices for the first stage, \(n\) is the number of choices for the second stage, \(p\) is the number of choices for the third stage, and so on.


Going back to the above example, use the Multiplicative Counting Principle to find how many possibilities of meal combinations there are.

Utilizing the following formula:

\(\text{Outcomes} = m * n\)

Where \(m\) is the number of choices for the first stage and \(n\) is the number of choices for the second, we get the following:

\(\text{Outcomes} = 2 * 2\)

\(\text{Outcomes} = 4\)

Therefore, using the Multiplicative Counting Principle, we can determine there are 4 possible meal options.

Indirect Counting Method

The indirect counting method is used to count the unwanted cases and subtract from the total possibilities.

We can determine this algebraically as such:

\(\text{Valid Outcomes} = \text{Total Outcomes} - \text{Unwanted Cases}\)


A standard die has 6 sides, numbered 1 to 6. Using the Indirect method, find out how many ways we can roll a number other than 3.

There are 6 possibilities when rolling a dice (1,2,3,4,5,6).

Landing on a 3 only happens when landing on the side of the dice that has the number 3.

We can calculate the number of valid outcomes as such:

\(\text{Valid Outcomes} = 6 - 1\)

\(\text{Valid Outcomes} = 5\)

Therefore, we can determine there are 5 valid outcomes for rolling a number other than 3.