Combinatorics: Introduction

Maria, Lina, and Mikael are going to the cinema. Each of them wants to sit in the middle. This will be difficult. In how many different ways the 3 friends can sit in 3 chairs? In other words, how many sequences are possible when the 3 friends take 3 chairs?

One way to solve this is to simply make a list of all possible sequences. When there are only 3 people and 3 chairs, it's quite easy to follow this way. But you can solve this problem a bit more systematically. Lets take it step by step. Anyone can sit in the first chair.

As for the second chair, we have only 2 friends left. And when the first 2 chairs are taken, there are only 1 person and 1 chair left, so there's only one choice. Each line on the tree diagram shows one possible sequence, or in other words, one permutation. The number of the n's at the bottom of the tree equals the number of permutations, or the total number of possible sequences. We solved the problem by using a diagram.

solve it in a numerically. Now we're going to Take a look. The first row has 3 choices. For each of those alternatives, there are 2 more choices. That's 3 x 2, In the third stage, we only have one choice left.

So we multiply by 1. You get to the number permutations by taking 3 x 2 x 1 You might see a pattern here. The number of permutations is given by the number of people multiplied by the number of people minus 1. multiplied by that number minus 1. And so on, all the way down to 1.

Had there been 5 friends going to the cinema, we would have calculated the number of possible permutations like this - 5 x 4 x 3 x 2 x 1 The row of factors can become very long. If you're calculating in how many ways 100 people can sit down, in 100 chairs, you'll fill the whole page. Therefore we use this symbol: ! We read it as the factorial. The factorial of 3! = 3 x 2 x 1 Or it is equal to 6.

And to the factorial of 4! = 4 x 3 x 2 x 1 Or 24. Check if your calculator has the factorial button. The factorial is a mathematical function. For a positive integer, it's factorial equals the product of all integers before it. Starting with 1.

Factorials are useful for calculating the number of possible sequences or permutations. No, this is not a permutation. Everyone has to sit in their own chair.