# What is a permutation calculation calculating?

##### 1 Answer

it's a calculation that shows the number of ways a group can be picked if we care what the order of those picks is (like the way election results for student government can go).

#### Explanation:

Let's say we have a student government and there are 4 seats to be filled - the President, the Vice President, the Treasurer, and the Secretary. Let's also say there are 4 people running for those seats. How many different ways are there for those 4 people to hold those 4 positions?

We can look at see that for the President's seat, there are 4 candidates.

We can then look at the VP's seat and see that, with someone in the role of President, there are now 3 people left who can be VP.

Then we look to the Treasurer and see there are 2 people left.

And lastly there is the Secretary and the remaining person goes there.

Therefore there are:

This is an example of a factoral problem - we're going to take a population and take all the members of that population and place them (whether it's political office or movie seats or whatever - we're using the entire population).

What happens if we have more population than we have seats or places? For example, there are 6 people who are vying for those 4 government slots?

We could do it this way:

We can look at see that for the President's seat, there are 6 candidates.

We can then look at the VP's seat and see that, with someone in the role of President, there are now 5 people left who can be VP.

Then we look to the Treasurer and see there are 4 people left.

And lastly there is the Secretary and there are 3 remaining people who could go there.

So we get

Notice that in the "6 people, 4 offices" example, we ended up with this calculation:

How can we put that in terms of factorials? Well, if we multiply it by

And keep in mind that the definition of a factorial is that we multiply by each natural number up to and including the one with the sign (so

And now see that the number 6 is the number of our population while 2 is the difference between the population and the number of seats that needed to be filled,

And that is exactly what the permutation formula is! If I call

Lastly, I want to show that this formula still works if we have that initial example - 4 people running for 4 seats - we should get