Counting

Counting is the concepts relating to the total possible number (i.e. count) of ways to do something given a certain population. The common forms include combination and permutation.

The Fundamental Principle

The fundamental principle of counting states that given k operations are performed, if there are $n_{i}$ ways to perform the i-th operation, then the total number of ways to perform the sequence of k operations is the product of all n. This makes sense so I'm not gonna elaborate.

Permutation

A permutation of a population is an ordering of a collection of objects. We calculate this with $n!$, and it makes sense so I'm not gonna elaborate.

Combinations

A combination of a population is a unique subset of it (i.e. a distinct group of objects). Notably different from permutations, the order does not matter.

The calculation is a bit complicated. Given $n$ items and $k$ items to insert, the total number of ways to select it can be calculated by the following:

$\left(^{n}_{k}\right)={\frac {n!}{(n-k)!k!}}$

Quick derivation: the numerator is the number of possible permutations, and the denominator is the number of ways to arrange a permutation. Since order don't matter, we remove the permutations with different ordering but the same observations.

For all this, just use a calculator.