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ZingPath: Permutations and Combinations

Permutations and Their Properties

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Permutations and Combinations

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Permutations and Their Properties


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Apply the permutation formula to solve problems.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Determine all possible arrangements using a tree diagram or list where the order is important.
  • Define what a permutation is.
  • Apply the permutation formula to solve problems.

Everything You'll Have Covered


A factorial is the product of all positive integers up to and including a given integer. It is denoted with the exclamation mark (!). The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.

For example,

In general, n! = n (n ? 1) (n ? 2) (n? 3)... (1)

0! has a special definition attached with it: 0! = 1


Permutations specifically count the number of different ways a task can be arranged or ordered.

An order of arrangements of r objects, without repetition, selected from n distinct objects is called a permutation of n objects taken r at a time, and is denoted as:

where n is the number of elements available for selection and r is the number of elements to be selected.

In other words, when you need to count the number of ways you can arrange items where order is important you can use permutation to count. You may want to know how many ways to pick a 1st, 2nd, and 3rd place winner from 10 contestants. Since you are arranging them in order, you could use a permutation to do this. Or if you wanted to know how many ways your committee could pick a president, a vice president, a secretary, and a treasurer, you could use permutations.

For example,

There are 20 runners in a race that will be ranked 1st, 2nd, and 3rd place as they cross the finish line. How many ways can the runners place?

Since we are ranking the runners, order is important, so we can use permutations to find the answer.

First, find n and r.

n is the number of runners in the race, which is 20.

r is the number of runners that will place in one race, which is 3.

Substitute these numbers into the permutation formula and simplify:

This means that there are 6,840 different ways the runners could place in the race.

Special Permutations

So, P(0, 0) = P(1, 0) = P(1, 1) = 1

Also, the number of selections of n distinct objects among n objects is n!.

Tutorial Details

Approximate Time 25 Minutes
Pre-requisite Concepts counting principle, factorial notation
Course Geometry
Type of Tutorial Concept Development
Key Vocabulary counting principle, factorial, permutation