# What exactly is a permutation?

I know this may seem like a trivial question, but I really didn't know where else to ask it. My textbook doesn't do a very good job of explaining how a permutation differs from a standard problem in which you are given multiple "slots" in which various items can be "deposited" and are asked how many ways this can be done.

For instance, the text provides the following example problem:

A wedding party consisting of a bride, a groom, two bridesmaids, and two groomsmen line up for a photo. How many ways are there for the wedding party to line up?

In total, there are 6 different people. I labeled them $A, B, C, D, E,$ and $F$ for convenience. There are also six slots in which these people can stand. The number of possible combinations, then, is $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$. This is simply by the generalized product rule for combinations.

Have I just been finding permutations all along in these "how many different ways can X be arranged" problems without knowing that's what I've been doing? Any clarification would be greatly appreciated, as I'm pretty confused.

• Yes, these are really just the same thing: permutations, arrangements, and 'putting different things into different slots' – Bram28 Jun 27 '17 at 17:43
• Thanks for your response. It boggles my mind as to why my text would only introduce this terminology 4 sections into the chapter on counting if that's what we've been doing all along :P – AleksandrH Jun 27 '17 at 17:46
• Mathematicians are not always the best educators ... – Bram28 Jun 27 '17 at 17:50

A permutation of an $n$-element set $X$ is a bijection $X\to X$.
If for instance $X=\{1,...,n\}$, then you can see the choice of a permutation of $X$ as a way of choosing, for each ball (numbered $1$ through $n$) a slot (numbered similarly) so that each slot has a ball in it, and precisely one.
So in a sense, counting permutations of $X$ is counting the number of ways that one can put the balls in those slots. Adding constraints on how the balls should be put in the slots is like adding constraints on the permutations. For instance you can add the constraint that no ball is to be deposited in the slot with the same number, and that constraint becomes, in terms of permutations: the permutation has no fixed point.