Counting Principles: Learn It 2

Finding the Number of Permutations of [latex]n[/latex] Distinct Objects Using the Multiplication Principle

The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.

To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply.

Suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.There are four options for the first place, so we write a [latex]4[/latex] on the first line.Four times two blanks spots.After the first place has been filled, there are three options for the second place so we write a [latex]3[/latex] on the second line.

Four times three times one blank spot.

After the second place has been filled, there are two options for the third place so we write a [latex]2[/latex] on the third line. Finally, we find the product.

There are [latex]24[/latex] possible permutations of the paintings.

How To: Given [latex]n[/latex] distinct options, determine how many permutations there are.

  1. Determine how many options there are for the first situation.
  2. Determine how many options are left for the second situation.
  3. Continue until all of the spots are filled.
  4. Multiply the numbers together.
At a swimming competition, nine swimmers compete in a race.

  1. How many ways can they place first, second, and third?
  2. How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)
  3. How many ways can all nine swimmers line up for a photo?

Finding the Number of Permutations of [latex]n[/latex] Distinct Objects Using a Formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula.

Before we learn the formula, let’s look at two common notations for permutations. If we have a set of [latex]n[/latex] objects and we want to choose [latex]r[/latex] objects from the set in order, we write [latex]P\left(n,r\right)[/latex]. Another way to write this is [latex]{}_{n}{P}_{r}[/latex], a notation commonly seen on computers and calculators. To calculate [latex]P\left(n,r\right)[/latex], we begin by finding [latex]n![/latex], the number of ways to line up all [latex]n[/latex] objects. We then divide by [latex]\left(n-r\right)![/latex] to cancel out the [latex]\left(n-r\right)[/latex] items that we do not wish to line up.

Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is [latex]6\times 5\times 4=120[/latex]. Using factorials, we get the same result.

[latex]\dfrac{6!}{3!}=\dfrac{6\cdot 5\cdot 4\cdot 3!}{3!}=6\cdot 5\cdot 4=120[/latex]

There are [latex]120[/latex] ways to select [latex]3[/latex] officers in order from a club with [latex]6[/latex] members. We refer to this as a permutation of [latex]6[/latex] taken [latex]3[/latex] at a time. The general formula is as follows.

[latex]P\left(n,r\right)=\dfrac{n!}{\left(n-r\right)!}[/latex]

Note that the formula stills works if we are choosing all [latex]n[/latex] objects and placing them in order. In that case we would be dividing by [latex]\left(n-n\right)![/latex] or [latex]0![/latex], which we said earlier is equal to [latex]1[/latex]. So the number of permutations of [latex]n[/latex] objects taken [latex]n[/latex] at a time is [latex]\frac{n!}{1}[/latex] or just [latex]n![/latex].

formula for permutations of [latex]n[/latex] distinct objects

Given [latex]n[/latex] distinct objects, the number of ways to select [latex]r[/latex] objects from the set in order can be calculated using the permutation of [latex]r[/latex] objects out of a set of [latex]n[/latex] objects.

 

The formula is:

[latex]P\left(n,r\right) = {}_{n}{P}_{r} =\dfrac{n!}{\left(n-r\right)!}[/latex]

A professor is creating an exam of [latex]9[/latex] questions from a test bank of [latex]12[/latex] questions. How many ways can she select and arrange the questions?