I am starting to study probability and I have a problem where you have to find all possible pairs that can be made by rolling two dice, e.g. $(1,2) = (2,1)$.
By symmetry, I'm pretty sure the answer is $21$, but I know there has to be a combinatorial way to get to the result which I haven't found out yet.
If somebody could help me, I would be glad!
Here is an pretty simple way to think about it (although the method in the comment will work).
Go done number by number, or rather choice by choice for the first number.
For 1, you have 6 possible choices for the second die all of which will generate a unique pair.
$$(1,1) (1,2) ... (1,6)$$
Then for the next number 2: you only have 5 choices that will generate a unique pair (because $(1,2) = (2,1)$ and $(1,2)$ is present above.
Then keep doing this until you hit 6 (or however many sides you have for your die) and then add the results. Which will be
$$6+5+4+3+2+1 = 21$$