Forming groups is an important part of **every day life**. Whether it be in your work environment, your class room, or even in your daily life, grouping people together to achieve a goal is imperative.

Group formation is an interesting study that looks into how people are grouped together and the effects of the group composition on the outcome of a task or goal.

How * many different groups* of n people can be formed if there are m^n members in the total population? This question is answered by group theory, a branch of algebra that studies how things are arranged or organized.

Group theory can be applied to many areas such as mathematics, chemistry, and computer science. It is used to **understand organizing systems** such as compounds or circuits. It has been proven very useful in solving problems with these systems.

This article will discuss how group theory applies to the question asked in this paragraph and how you can use it to find how many different groups of n people can be formed if there are m^n members in the total population.

## Group of 7 people and 1 empty position

Now let’s go back to the group of 10 people, but this time, there is only **1 empty position**.

There are many ways to arrange the group of 7 people and 1 empty position. For example, one person can take the empty position, or several people can take the *empty position together*.

There are 2!^7 possible groups of 7 people and 1 empty position. This is because there are 2! ways to arrange the groups of 7 people, and there is only 1 empty position.

2! stands for 2 factorial, which is the number of **possible group arrangements multiplied** by itself one more time. Factorial stands for counting down from the highest number in a set to 0. For example: 5! = 5*4*3*2*1 = 120.

## How to calculate the number of different committees?

First, you have to figure out how **many different committees** can be formed from the total number of people in the group. To do this, you have to figure out how

*many different groups*of people you can put together.

There are 10 people in the group, so you can **form 10 groups** of people, each with one person. That makes it 10*1=10 committees that can be formed.

Then, to find the number of different committees that can be formed from the whole group, you have to combine all the groups that contain only one person. There will be 10 of these groups, so the total number of committees is 10*10=100.

## First committee

The first committee can be formed by choosing one person from the group and then choosing two people from the remaining nine people. The **two people must** then *choose one person* each to form the committee of seven members.

This is a very powerful committee since there are two leaders and five members. The leaders can work together or against each other, depending on their personal relationships.

The strength of this committee is that there are two leaders, so if one leader is voted out, then there is still a **strong leader remaining**. The weakness of this committee is that there are only five members, so if more issues need to be addressed, more people may need to be invited to address them.

Second committee

The second type of committee that can be formed from the group of ten people is a very small one with only three members. These *three members must choose two* more people to form a group of seven members.

## Second committee

Now let’s look at the second way to form a committee — by choosing people based on their relationships with other people in the group.

You can assume that people in a group know each other fairly well, so you can probably guess who knows whom in the group.

If you ** pick two close friends** as committee members, then you can be pretty sure that they are friendly with one another, which means they have someone in common — the other committee member.

So if you pick two close friends as committee members, and one of them has two close friends of his or her own, then you can pick one of those friends to be on the committee.

If none of the members of the group are close friends, then you can *still pick one person* to be on the committee — just not someone who is totally unacquainted with anyone else in the group.

## Third committee

Now let’s look at the third possible committee. We will call this the rotation committee. This committee requires all members of the first committee to ** rotate one position** to the left, and all members of the second committee to rotate one position to the left.

So in other words, you would take all of the A positions and make them B positions, then take all of the B positions and make them A positions. Then you would take all of the C positions and make them A positions, then take all of the D positions and make them B positions.

This would result in a **new first committee made** up of people from the original A and B committees, and a **new second committee made** up of people from the original C and D committees. Your total number of committees remains at three!

This is an important concept to understand when trying to figure out how many different committees can be formed from a group of 10 people. It does not depend on how many people are in each individual committee.

## Fourth committee

Now let’s look at the **fourth possible committee**. In this case, the group of *10 people consists* of 4 A’s and 6 B’s.

Since there are 4 A’s, we can **form 1 committee** of all A’s. Since there are 6 B’s, we can form 1 committee of all B’s.

We can also form a committee of all A’s and B’s. How? First, put all the A’s in one group and then put all the B’s in another group. Then, form a committee consisting of all the A’s and another committee consisting of all the B’s.

You *could also form 2 committees*: One with 5 A’s and 5 B’s, or one with 4 A’s and 8 B’s.

## Fifth committee

So, the fifth committee can be formed by choosing five people from the group and assigning them to work together. However, this committee can only *include three people* from the original group, as two people from the original group must be replaced.

Two of the **original members must** be replaced by two new members from the larger group. This is because you need to have five people in the committee, and you need to have all *three departments represented*.

The remaining member of **one department must join** with two members from another department to form the fifth committee. It does not matter which two departments they are from, they must work together to form this new committee.

Fourth committee

The fourth committee can be formed by choosing four people from the larger group and assigning them to work together. Any four members of the larger group can form this committee, as long as they all work together.

## Sixth committee

Now, let’s look at the sixth possible committee of seven. In this case, the group of *ten people contains three* very close friends and two strangers. The three very close friends all know each other, and the two strangers do not know each other.

We will call the three very close friends group A, and the two strangers group B. As we have seen before, group A can form a committee with group B, and they can form a committee with themselves, so that *makes two possible committees*.

Group A cannot form a committee with group B or themselves if one of them does not know each other. As we have also seen before, if all of them know each other then they can all form a committee together. So there is one more possible committee than before!

Group B cannot form a committee with themselves or any other groups because none of them know each other.