how many different ways can five people be lined up?

Question

how many different ways can five people be lined up?

Answer 1

Hello, thank you for taking the time to post your question!

Since the order matters in this case the number of ways to arrange “n” items is given by the factorial of “n” … here n = 5, meaning that

5! = 5 x 4 x 3 x 2 x 1 = 120

So there are 120 different ways to line up the five people

I hope that helps you get moving in a better direction on this type of question! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

This is a permutation problem with a standard answer.  If the number of people is n, then the answer is n! (n factorial).  So in this case the answer is 5! = 120

 

To see how it works, first look at how many different choices there are for the first person.  It could be any of the 5 people.

 

Now, how many choices are there for the second person?  Since there are only 4 left, there are only 4 people who could be chosen.  So the total number of different orderings for the 1st and 2nd people combined would be 5*4 = 20.

 

Now look at the third person.  There are only three people left, so there are 3 choices.  So the total number of different orderings for the 1st, 2nd and 3rd people combined would be 5*4*3 = 60.

 

You continue that way until the last person.  So no matter how many people you start with, the number will be n * (n-1) * (n-2) * (n-3) * ... * 1, which is equal to n!.