When it comes to the formula for $r$-combinations, I was initially unsure of how to derive it myself, notwithstanding a big portion of the formula being the permutation formula which was fairly easy to grasp. It was the factor of the denominator, $r!$, that I wanted to understand.

Here's the formula: $${n \choose r}= \frac{n!}{r!(n-r)!}$$

To figure out how we got this formula, we must ask ourselves how we can derive it from an earlier principle: the product rule.

Well it turns out we can, and it's about as easy to grasp as permutations.

Firstly, it's helpful to define the difference between unsorted and sorted sequences.

unsorted sequence: an $r$-combination sorted sequences: an $r$-permutation