9.6 Counting

9.6: Counting

Counting is a very useful skill that can be applied to solving probability problems. Counting allows you to find out the number of possible ways something can occur.

The simplest counting problems can be solved by listing out all of the possible outcomes of a given situation. For example, you are given 8 pieces of paper numbered 1 to 8 and asked how many combinations of two papers will add up to 12. To solve this problem, you could simple list out all of the different combinations of numbers between 1 and 8 that add up to twelve. By making a table like the one below, you can count that there are 5 different ways of obtaining a sum of 12.

First Number	4	5	6	7	8
Second Number	8	7	6	5	4

However, most problems aren’t this simple. When a problem has hundreds or thousands of different outcomes, writing out all of the possible outcomes becomes an issue.

One way of solving such problems is to use the Fundamental Counting Principle.

Put simply, the fundamental counting principle states that the number of ways that two events can occur is (Number of ways the first event can occur) x (number of ways the second event can occur, after the first event has already occurred), or (m₁) (m₂).

The fundamental counting principle can also be applied to more than just two events. For example, you could have (m₁)(m₂)(m₃)

Sample problem: How many different combinations of letters and numbers are there for license plates if they always follow the pattern of LNNNLL? (L = letter, N=number)

Using the fundamental counting principle, you simply multiply together the amount of ways each of the 6 characters on the license plate can occur. Since there are 26 letters in the alphabet, and 9 digits, you can find the answer by multiplying (26)(9)(9)(9)(26)(26).

A Permutation is an ordering of elements where such that order matters.  For example, in a permutation, ABC is considered a different combination than CBA.

The equation for finding the number of permutations of n elements is simply n!

Sample problem: How many permutations are there for the letters A, B, C, D, E, and F?

To solve this, you simply do 6! because there are 6 elements. To help better understand this, think of it step by step. For the first element, you have 6 choices. Then for the second one, you have 5 choices since on of your elements was used up in the first step. Then for the third you have 4 choices, and so on.

Finding the number of permutations is not always so simple. Sometimes, you have to order subsets of elements, rather than just ordering one element at a time. You can do this by using the equation for _nP_r, which simply mean n elements chosen r at a time. The equation for this is _nP_r =

Sample Problem: In a race of 12 people, how many ways can these people come in first, second, and third?

To solve this, you must first identify that you are choosing from 12 people, 3 at a time (first, second, and third). So, you know that you can replace _nP_r with ₁₂P₃, which you can then plug in. You can either solve this using the equation, or type it into your calculator.

Sometimes, you will need to find the number of distinguishable permutations. You need to do this when you have elements that are the same. The equation for this is

What you are dividing by is the amount of elements that are the same to eliminate identical permutations. This can be easily demonstrated in the next sample problem.

Sample Problem: How many distinguishable ways can the letters of WISCONSIN by written?

All you do to solve this is plug it into the above equation so you get

In this problem, n=9 because WISCONSIN has 9 letters. On the bottom, you have 2! once because there are two I’s in the word, and again because there are two S’s in the word. So, n₁ and n₂ and so on are just the number of times a certain element is repeated. 

A Combination is the ordering of elements when order DOESN’T matter. So, ABC would be considered the same as CBA.

When looking for the number of combinations of n elements, the approach is similar to finding permutations. To begin, you plug information into _nC_r, or n elements chosen r at a time. From there, you simply plug n and r into the equation _nC_r =

If you have a calculator, you can just type in _nC_r to find the number of combinations. This operation can be found in the probability section of the MATH key.

Sample problem: How many different ways can 4 digits be chosen if the order of the digits doesn’t matter?

Since you have a total of 9 digits, which you are choosing 4 at a time, you can come up with the equation for ₉C₄. You simply plug this in to find your answer.

So, counting really isn’t that hard you just have to think and know which equation to use. Plus your calculator can help you a ton. So yeah. That’s counting.

-Corin