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# Permutation Problems

## Using the nPr function found in the Math menu under PRB  on the TI calculator

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Practice Permutation Problems
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Evaluate Permutations Using Permutation Notation

Do you know how to calculate a permutation in permutation notation? Take a look at this dilemma.

Find 8P2\begin{align*} _{8}P _{2}\end{align*}

To figure out this permutation, you will need to understand permutation notation. Pay attention and you will learn all that you need to know in this Concept.

### Guidance

Order is important in some situations and not important in others. For example, in following a cake recipe, the order in which the events take place is important. You need to crack the eggs before you mix them with the flour. Similarly, you put the icing on the cake only after it has baked.

In buying the ingredients to make a cake, on the other hand, order is not important. Does it matter if you buy the flour before the eggs or the milk before the icing? It doesn’t, so you would say that order is NOT important in buying cake ingredients.

For solving many problems in which order is important, you can use permutations. A permutation is a selection of items in which order is important. To use permutations to solve problems, you need to be able to identify the problems in which order, or the arrangement of items, matters.

We can find the number of permutations in a group if you include all members of that group. For example, suppose there are 3 cabs in front of a hotel, Acme, Bluebird, and Checker. If all 3 line up to wait for the next customer, the number of different lineups, or permutations, of 3 items taken 3 at a time is:

Again, this is the permutation for three cabs lined up three at a time. We could also say that this is three objects taken three at a time.

The most efficient way to calculate permutations uses numbers called factorials.

Factorials are special numbers that represent the product of a series of descending numbers.

The symbol for a factorial is an exclamation sign. Take a look.

8!11!4!17!=8factorial=87654321=11factorial=1110987654321=4factorial=4321=17factorial=1716151413121110987654321

To compute the values of factorials, simply multiply the series of numbers.

4!5!8!11!=4321=24=54321=120=87654321=40,320=1110987654321=39,916,800

We can use factorials to calculate permutations.

Suppose you have 6 items and you want to know how many arrangements you can make with 4 of the items.

Order matters in this problem, so you need to find the number of permutations there are in 6 items taken 4 at a time.In permutation notation you write the following.

6P4 \begin{align*}{\color{red}_{6}}P {\color{blue}_{4}} \ \Longleftarrow \end{align*} 6 items taken 4 at a time

In general, permutations are written as:

nPr n\begin{align*}{\color{red}_{n}}P {\color{blue}_{r}} \ \Longleftarrow {\color{red}n}\end{align*} items taken r\begin{align*}{\color{blue}r}\end{align*} at a time

To compute nPr\begin{align*}{_n}P{_r}\end{align*} you write:

nPr=n!(nr)!=total items! (total itemsitems taken at a time)!

To compute 6P4\begin{align*}_{6}P _{4}\end{align*} just fill in the numbers:

6P4=6!(64)!

7202\begin{align*}\frac{720}{2}\end{align*}

360\begin{align*}360\end{align*}

Notice that it is the product of the values in descending order that tells us how many permutations are possible.

You can use this method to solve any number of permutations.

Find each permutation.

#### Example A

Find 4P3\begin{align*} _{4}P _{3}\end{align*}

Solution: 24\begin{align*}24\end{align*} options

#### Example B

Find 12P2\begin{align*} _{12}P _{2}\end{align*}

Solution: 132\begin{align*}132\end{align*} options

#### Example C

Find 8P6\begin{align*} _{8}P _{6}\end{align*}

Solution: 20,160\begin{align*}20,160\end{align*} options

Now let's go back to the dilemma from the beginning of the Concept.

Find 8P2\begin{align*} _{8}P _{2}\end{align*}

This is the same as saying 8 options taken 2 at a time.

We can multiply to solve the permutation.

8×7=56\begin{align*}8 \times 7 = 56\end{align*}

There are 56 options.

### Vocabulary

Permutation
a selection of items in which order is important.
Factorial
a special number that represents the product of a series of descending values.

### Guided Practice

Here is one for you to try on your own.

Find 7P3\begin{align*} _{7}P _{3}\end{align*}

Solution

Step 1: Understand what 7P3\begin{align*} _{7}P _{3}\end{align*} means.

7P3 \begin{align*} {\color{red}_{7}}P {\color{blue}_{3}} \ \Longleftarrow \end{align*} 7 items taken 3 at a time

Step 2: Set up the problem.

7P3=7!(73)!=total items!(total itemsitems taken at a time)!

Step 3: Fill in the numbers and simplify.

7P3=7!(73)!=7!4!=7×6×5×4×3×2×14×3×2×1=7×6×51=210

There are 210 possible permutations.

### Practice

DirectionsFind each permutation.

1. Find 7P2\begin{align*} _{7}P _{2}\end{align*}
2. Find 6P3\begin{align*} _{6}P _{3}\end{align*}
3. Find 5P4\begin{align*} _{5}P _{4}\end{align*}
4. Find 5P5\begin{align*} _{5}P _{5}\end{align*}
5. Find 9P3\begin{align*} _{9}P _{3}\end{align*}
6. Find 9P7\begin{align*} _{9}P _{7}\end{align*}
7. Find 11P3\begin{align*} _{11}P _{3}\end{align*}
8. Find 12P3\begin{align*} _{12}P _{3}\end{align*}
9. Find 6P2\begin{align*} _{6}P _{2}\end{align*}
10. Find 14P3\begin{align*} _{14}P _{3}\end{align*}
11. Find 15P3\begin{align*} _{15}P _{3}\end{align*}
12. Find 11P4\begin{align*} _{11}P _{4}\end{align*}
13. Find 16P2\begin{align*} _{16}P _{2}\end{align*}

Directions: Use permutations to solve each problem.

1. Mia has 7 charms for her charm bracelet – a heart, a moon, a turtle, a cube, a bird, a hoop, and a car. Into how many different orders can she arrange the 7 charms?
2. One of the charms in Mia’s bracelet in problem 6 above fell off. How many fewer arrangements are there now?

### Vocabulary Language: English

combination

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.
factorial

factorial

The factorial of a whole number $n$ is the product of the positive integers from 1 to $n$. The symbol "!" denotes factorial. $n! = 1 \cdot 2 \cdot 3 \cdot 4...\cdot (n-1) \cdot n$.
n value

n value

When calculating permutations with the TI calculator, the n value is the number of objects from which you are choosing.
Permutation

Permutation

A permutation is an arrangement of objects where order is important.
r value

r value

When calculating permutations with the TI calculator, the r value is the number of objects chosen.