Introduction to Word Problems with the Counting Principle, Permutations, and Combinations
The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.
—JOHN KEMENY (1926–1992)
This lesson will cover the counting principle, permutations, and combinations. The steps to solving word problems involving these topics will be detailed.
The Counting Principle
The counting principle states that the total number of possibilities can be found by multiplying the number of choices.
In addition to this strategy, another way to find the total number of possibilities is to take the number of choices in each category and multiply them.
Take the following example.
Sean has three ways to get to school. He can take the bus, get a ride, or walk. If he can take one of the same three ways to get home, how many total choices does he have to get from home to school and back?
Read and understand the question. This question is looking for the total number of possibilities when there are three different modes of transportation to school and back.
Make a plan. The trip from home to school is one route he needs to take, and the trip from school back to his home is the second route he needs to take. Take the total number of choices for each route and multiply them.
Carry out the plan. There are three different ways to get from home to school, and three different ways to get from school back to home. Multiply these choices: 3 × 3 = 9 different ways.
Check your answer. To check this answer, divide the total number of outcomes by the number of choices in the first route. This becomes 9 ÷ 3 = 3, which is the number of choices in the second route. This answer is checking.
Permutations
A permutation of objects is the total number of different orders of the objects. In other words, it is the number of arrangements that can be made when you are working with a certain set of objects.
The number of permutations of a set of objects is based on the counting principle, so take the number of choices for each placement in the set and multiply them. This will give the total number of permutations of the set.
For example, find the total number of permutations (orders) of the set of letters A, C, and T. The permutations do not have to form regular words.
Read and understand the question. This question is looking for the total number of permutations, or orders, of three different letters. Each letter is used exactly once.
Make a plan. Multiply the number of choices for each placement of the letters. A letter can only be used once.
Carry out the plan. There are three letters to choose from for the first letter, two letters to choose from for the second letter, and thus only one letter to choose from for the third letter. The number of permutations is therefore 3 × 2 × 1 = 6.
Check your answer. One way to check this solution is to use the strategy of making an organized list. The possible orders of the letters are ACT, ATC, CAT, CTA, TAC, and TCA. There are 6 different orders, so this solution is checking.
Tip:
When multiplying the number of choices to find the permutations of the objects, the number of choices always starts with the total number of objects and decreases by one each time. For example, when you are finding the number of permutations of 5 objects taken 5 at a time, multiply 5 × 4 × 3 × 2 × 1 = 120 different permutations.
However, sometimes not all of the objects are used in each permutation. For instance, when you are finding the number of permutations of 5 objects taken only 3 at a time, start with 5 choices and multiply just the first 3 values together. The number of permutations in this situation is 5 × 4 × 3 = 60.

Combinations
The major difference between the combinations of a set of objects and the permutations of the same set of objects is that with combinations, the order does not matter. When the order of the objects changes, it is still considered the same combination of items.
Calculating the total number of combinations is very similar to finding the number of permutations. However, because the order does not matter, you must divide by the total number of different orders of the objects.
Tip:
Because the order does not matter, the total number of combinations of n objects taken n at a time is always equal to 1. For example, the total number of combinations of 3 objects when all 3 objects are used is 1. Even though you can list the objects in different orders, the set still contains the same 3 objects.

Take the next example.
How many different combinations of 3 students can be selected for a committee when there are a total of 10 students from which to choose?
Read and understand the question. This question is looking for the total number of combinations of 10 students selected 3 at a time for a committee.
Make a plan. Because the order is not important, find the number of permutations of 10 students taken 3 at a time, and then divide by the number of permutations of 3 students taken 3 at a time.
Carry out the plan. The number of permutations of 10 students taken 3 at a time is equal to 10 × 9 × 8 and the total number of permutations of 3 students taken 3 at a time is equal to 3 × 2 × 1. Divide the number of permutations of 10 students taken 3 at a time by the number of permutations of 3 students taken 3 at a time to find the total number of combinations of 10 students taken 3 at a time.
different combinations.
Check your answer. To check this solution, find the number of combinations of 10 students taken 7 at a time, which should be the same result. To do this, calculate the number of permutations of 10 students taken 7 at a time and divide this amount by the permuations of 7 students taken 7 at a time. The number of combinations is equal to
so this solution is checking.
Tip:
When you are looking at the total number of permutations of a set of objects, the order matters. When looking at the total number of combinations of a set of objects, the order does not matter.

Find practice problems and solutions for these concepts at Word Problems with the Counting Principle, Permutations and Combinations Practice Questions.
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