**High School Number Sense Lesson 51: Permutations**

We use

**permutations**as a real-life application of

**factorials**, which we covered a bit in

**FactorialOper**. Problems with permutations are often disguised as word problems, but we will cover some of each type found on number sense tests in this lesson. This concept appeared

**8 times**last year, with the median placement at

**question # 56**.

**Number Dojo Level: 250**

By definition, a

**permutation**is an ordering of a set of numbers or object into some sequence. They differ from

**combinations**(which we'll cover later this week), where order is disregarded. The number of permutations of

**n**distinct objects is

**n factorial**, written as

**n!**, meaning the product of all positive integers less than or equal to n.

For example, the number of permutations of a set of 3 objects {a, b, c} is 3!, and they are:

- abc
- acb
- bac
- bca
- cab
- cba

**Application**

A common question on recent number sense tests has been "How many ways can 4 books be arranged on a shelf?" The answer to this question is

**4!**, which is 4 x 3 x 2 x 1 =

**24**.

**Example 1: The number of ways to arrange five people in a line is ___**

- The answer is 5!, which is
**120**.

**Example 2: 6 books can be arranged on a shelf ___ different ways**

- The answer is 6!, which is
**720**

Variation

Variation

Often, permutation problems are represented as "

**n**objects taken

**k**at a time," which is written as:

_{n}P_{k}**n!**by

**(n - k)!**, or more visibly:

**n!**

----------

(n - k)!

----------

(n - k)!

**Example 3: The number of permutations of 5 items taken 2 at a time is ___**

- This will be calculated as
**5!/(5 - 2)!**or**5!/3!** - Think of this as
**(5 x 4 x 3!)/3!**Notice that the**3!**cancels from the top and bottom. - The answer is 5 x 4 =
**20**.

**Example 4: The number of permutations of 6 items taken 4 at a time is ___**

- This will be calculated as
**6!/(6 - 4)!**or**6!/2!** - Think of this as
**(6 x 5 x 4 x 3 x 2!)/2!**Notice that the**2!**cancels from the top and bottom. - The answer is 6 x 5 x 4 x 3 =
**360**.

**Distinguishable Permutations**

If repetition of objects is allowed, we need another formula. Let's say that we are making real or imaginary anagrams of the word

**circle**. Notice that

**c**is listed twice. The formula for this is

**6!/2!**because there are 6 letters with one letter repeating twice. The answer is the same as Example 4:

**360**.

But what if more than one letter repeats, as in

**onion**? The formula would be

**5!/(2!2!)**, which is

**120/(2 x 2)**, which is

**30**.

**Example 5: How many words, real or imaginary, can be made using the letters in the set {b, e, i, g, e}?**

- There are
**5**letters, and**e**repeats twice. The formula is**5!/2!** - The answer is 120/2 =
**60**.

**Example 6: How many words, real or imaginary, can be made using the letters in the word KANSAS?**

- There are
**6**letters, with A repeating 2 times and S repeating 2 times. The formula is**6!/(2!2!)**. - This can be simplified as
**(6 x 5 x 4 x 3 x 2!)/(2! x 2!)**. Notice the**2!**cancels from the top and bottom. - The answer is (6 x 5 x 4 x 3)/2 = 360/2 =
**180**.

**Here's a free worksheet to help you practice Permutation:**

permutation.pdf |

**Up Next for High School: Combination**