Word Problems with the Counting Principle, Permutations, and Combinations Study Guide
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.
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.
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.