**Permutation and combination** are possible ways of arranging and selecting elements from a group of elements without possible replacement. Both topics are very important in mathematics. They explain the various ways of arranging some sets of data.

Permutation and combination along with the differences between them will be properly explained in this article. In addition, keynotes with real-life examples will be discussed. As a bonus, students will watch a video lesson of simplified explanations of two questions on permutation and combinations to enhance their knowledge and heighten their boldness in solving future permutation and combination questions.

###### WHAT IS PERMUTATION

Permutation occurs almost in all facets of mathematics. Since it has to do with the arrangement of objects from a group of elements. So permutation is defined as the arrangement of objects or numbers in a precise order. The issue of permutation often arises when items like (balls, boxes, books, etc.) are to be arranged in a definite order. And it involves an arrangement in which the “**order**” is important.

###### PERMUTATION FORMULA

We said earlier that permutation is the arrangement of elements in a specified order. So the arrangement of** r **items from a set of **n** objects in a precise order can be written as

Where the number of items and how many items are taken at a time.

###### PERMUTATION OF IDENTICAL OBJECTS

In most cases, permutation includes the problems of arranging objects that are repeated. For example in the word AFRICA, we may be required to find the number of ways of arranging the letters of the word AFRICA. In this arrangement, the two A’s looks alike. Writing the two A’s with different kinds of letters such as AFRICa, will show that we have 6! ways of arranging the word.

The two A’s have ways of arrangement without altering the other letters i.e. AFRICa or aFRICA. So the word AFRICA can be arranged in identical A’s ways, i.e.

∴

Furthermore, we can be given the word **MINIMUM**, and the asked to find the number of ways it can be arranged. In that word “MINIMUM” there are three M’s and two I’s, so the number of ways of arranging the 7 letter word such that three M’s and two I’s are identical is ways.

So

∴ ways

In general, the number of ways of arranging n objects of which r objects are identical is . Where

numbers of ways of arranging all n objects if they are different and

number of ways of arranging the identical objects.

###### WHAT IS COMBINATION

A combination is a mathematical technique of selecting objects or numbers from a group of objects or collections in such a way that the order of the selection does not matter. Note that combination involves “selection” in which the “**order**” of selection does not matter.

###### COMBINATION FORMULA

The combination is a selection of **r** elements from a set of **n** elements in which the order of selection does not matter. So the selection of **r** elements from a set of** n** elements can be written as

Where, number of elements and elements taken at a time.

###### DIFFERENCE BETWEEN PERMUTATION AND COMBINATION

PERMUTATIONS | COMBINATIONS |

Permutation involves arrangements | Combination involves selections |

Order is very important. For example, ab and ba are different arrangements | Order is not important. For example, ab, and ba are the same selection or grouping |

The formula for permutation is as above | The formula for Combination is as above |

**Note**: The main difference between permutation and Combination is the **ordering**. With permutation, more attention is given to the order of the elements (objects or numbers). Whereas with combination, we don’t care much about the order. For Example, if you lock your locker or bag with a number key or lock “combo” with the number 7654, and try 4675 to open the locker, it won’t open. This is because it is a different number ordering, this is a permutation.

The equation connecting Combination and Permutation is

###### FACTORIALS

A product such as is called factorial , and it is written as So factorial i.e. . Hence, the number of arrangements or permutation of n different object is . And the number of arrangement of two objects taken from n different ones is . Similarly and etc.

So the number of arrangements of objects taken from different ones is or . For example,

### Permutation and Combination Video Lessons 1

### Permutation and Combination Video lesson 2

For more insights on these and more topics in Mathematics, download our e-books. These e-books are sure-best for your success, contain all topics in your syllabus plus past questions. Click **HERE** to download a copy Now!

Let us consider some practical examples on these topic and the solutions. This will consolidate all you have learnt in this mathematics article.

###### PROBLEMS ON PERMUTATION AND COMBINATION

**Example 1: In how many ways can four books be arranged in order if eight different books are available.**

**Solution:**

The first Shelf can be filled by one of the eight books | 8 Ways |

The Second Shelf is filled by one of the seven books left | 7 Ways |

The third Shelf is filled by one of the six remaining books left | 6 Ways |

The fourth Shelf is filled by one of the five remaining books left | 5 ways |

Hence, the number of arrangement is

**Example 2**: If ,** find the value** **of**

**Solution:** will give us

If

Cross multiplying the above expression will give

.

So

,

which can still be written as

∴

,

.

Finally,

So

**Example 3**: A Committee of two men and three women is to be chosen from five men and four women. How many different committees can be formed?

**Solution: **The two men can be selected in ways.

Recall that generally,

So ways

Note that

Women can be chosen in

ways

Also note that

So therefore, there are possible committees.

Conclusively, having seen the above examples, I want you to try the questions on permutation and combination below. If you do care to join our mathematics forum or community, were you will want to post your solutions to the questions, you are most welcomed.

###### PERMUTATION AND COMBINATION QUESTIONS

Try the three questions below. After which you may decide to check more questions on this topic from other sources.

- In how many ways can we arrange the word MATHEMATICS.
- In how many ways can the 1st 2nd and 3rd prizes be awarded in a race if there are 10 competitors.
- Find the number of ways of solving 8 subjects from 12 subjects for an examination.

**Join our over 20,000+ readers to receive articles on current events in Nigeria and globally. Do you desire to get informed on Website design, Digital Marketing, External Examinations in Nigeria, and lots more? Visit our blog page to read articles on these and mor**e.