# Why are permutations important?

See below on some thoughts:

#### Explanation:

Let's first talk about what a permutation is. To do that, I'll first talk about factorials.

When we order a bunch of things and order is important (such as the number of ways to order the books in a 10 volume encyclopedia set), we can see that there are 10! ways to arrange the books - the first book on the shelf can be any of 10 books, the second on the shelf can be any of the 9 remaining, the third on the shelf can be any of the 8 remaining, and so on, giving:

10xx9xx8xx7xx6xx5xx4xx3xx2xx1=10! = 3,628,800

And this works great if we want to arrange everything you have on hand. But what if we want to arrange things but not all the things? Let's say we have 10 action figures but only have space on the shelf for 6 of them. How many different ways can we display the figures?

We could calculate it by saying that there are 10 figures we could put in position one on the shelf, then 9 in position two, 8 in position three, and so on, giving:

$10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 = \text{a lot of hitting the times key on the calculator}$

We can cut down this work by seeing that our multiplication string is the same as:

((10xx9xx8xx7xx6xx5)(4xx3xx2xx1))/(4xx3xx2xx1)=(10!)/(4!)

which we can rewrite:

(10!)/(4!)=(10!)/((10-6)!)

and now we have everything in terms of what we knew (picking 6 things from a population of 10 things) and this is what a permutation is:

P_(n,k)=(n!)/((n-k)!); n="population", k="picks"

A factorial is a set number - we know that 10! = 3,628,800 and 4! = 24, and so we can find that final answer by saying:

(10!)/(4!)=(10!)/((10-6)!)=3628800/24=151,200

So we've figured out that permutations are great for saving a lot of work when calculating the number of ways things can be ordered where the order of the arrangements is important. How much work? Let's consider this question:

"An airplane flight is oversold. There are 300 people holding tickets to get onto a plane that has 250 seats. How many different ways can we arrange people on the plane?"

The answer is P_(300,250)=(300!)/(50!)

(the approximate numerical answer is $9.5 \times {10}^{121}$)