TITLE: I'm studying beginner combinatorics. How do I know which one to use?
QUESTION [6 upvotes]: I'm studying combinations and permutations in high school. How do I know exactly when to use which? On tests, my teacher expects me to know when to use permutations and when to use combinations. I've read all over that if "order matters", use permutations, and if "order doesn't matter", use combinations, but that seems vague and prone to error.
I've done questions where it seems as though combinations are to be used since order doesn't matter, but permutations are to be used instead. I just don't understand when to use which.
Could someone explain when to use which and how to develop intuition for these problems? I get that I should just do them and develop an intuition, but I will be given unusual, never seen before problems on tests and I don't want a 50/50 chance of solving the problem.
Thanks.

REPLY [2 votes]: Use combinations when order doesn't matter. Use permutations when order matters.
Alternatively derive them on the spot.

Rule of product:
A decision with $m$ outcomes and another with $n$ outcomes regardless of which option was taken for the first has $mn$ outcomes.


How many ways are there to arrange 4 things chosen from a set of 10?

For the first thing we have 10 options, for the second we have 9 left, for the third we have 8, for the 4th we have 7, so the answer is:
$10*9*8*7=\frac{10!}{6!}=\ ^{10}P_4$


How many ways are there to choose 4 things from a set of 10.

There are $\frac{10!}{6!}$ ways to arrange them. But say we're given 4 things, there are 4 ways to choose the first, 3 for the second... so that means given 4 things there are $4*3*2*1$ ways to actually arrange them.
So each of our $\frac{10!}{6!}$ ways will be repeated $4!$  times (as these are all distinct arrangements) dividing by this we get: $\frac{10!}{6!4!}$ which you'll note is $\ ^{10}C_4$