Counting becomes more structured when we select several objects from a larger collection. The key question is whether the order of the selected objects matters.

Permutations: order matters

Suppose $k$ distinct objects are chosen from $n$ distinct objects and arranged in order. There are $n$ choices for the first position, $n-1$ for the second, and so on. The number of ordered selections is

$$P(n,k)=\frac{n!}{(n-k)!}.$$

If all $n$ objects are arranged, this becomes

$$P(n,n)=n!.$$

For example, choosing gold, silver and bronze medalists from $8$ competitors gives

$$P(8,3)=8\cdot7\cdot6=336.$$

Combinations: order does not matter

If only the selected group matters, every group of $k$ objects appears in $k!$ different orders among the permutations. Dividing those duplicate orderings gives

$$\binom nk=\frac{n!}{k!(n-k)!}.$$

For example, choosing a committee of $3$ people from $8$ gives

$$\binom83=56.$$

Choosing the model

A password, ranking or seating arrangement usually depends on order and leads to permutations. A committee, hand of cards or subset usually does not and leads to combinations.

The formulas are consequences of the multiplication principle; the main skill is identifying what counts as a genuinely different outcome.