Probability runs on counting, and counting runs on one question: does order matter? Answer that and the formula picks itself.

Arrange everything and you get a factorial: $n! = n \times (n-1) \times \cdots \times 2 \times 1$, read n factorial, where $n$ is the number of distinct items being placed. Five books on a shelf go $5! = 120$ ways: five choices for the first slot, four for the next, and the cascade runs down to one. Take only $r$ of the $n$ and keep the order, and the cascade simply stops early: ${}_{n}P_{r} = \dfrac{n!}{(n-r)!}$, a permutation, read n P r. Here $n$ is the size of the pool and $r$ is how many are chosen; gold, silver and bronze from ten sprinters is ${}_{10}P_{3} = 720$.

Drop the order and every selection has been counted $r!$ times over — once per reshuffle — so divide it back out: ${}_{n}C_{r} = \dfrac{n!}{r! \, (n-r)!}$, a combination, read n choose r. A committee of three from eight is ${}_{8}C_{3} = 56$, not 336. And a free identity worth remembering: ${}_{n}C_{r} = {}_{n}C_{n-r}$ — choosing who goes is the same act as choosing who stays behind.

The exam tell is in the noun. A slate, a podium, a ranking, a sequence: order matters, use nPr. A committee, a hand, a set, a pair: order does not, use nCr. Getting that one word right is worth more marks than any arithmetic in this lesson.