Relationship Between Permutations and Combinations

Sections 3 and 5 derived $^{n}P_{r}$ and $^{n}C_{r}$ independently, but they are not two unrelated formulas — $^{n}C_{r}$ was, in fact, derived directly from $^{n}P_{r}$ by dividing out the internal orderings. This section makes that connection explicit and shows how to decide which formula a given problem actually calls for.

The central relationship

The reasoning behind it, restated from Section 5: any ordered arrangement of $r$ objects out of $n$ can always be split into two independent stages — select the $r$ objects (in $^{n}C_{r}$ ways), then arrange the selected $r$ objects among themselves (in $r!$ ways). Because these are two stages of one combined process, the Fundamental Principle of Counting (Section 1) multiplies them together, giving $^{n}P_{r}$ as the total count of ordered arrangements. This is precisely why $^{n}P_{r}$ is always $r!$ times as large as $^{n}C_{r}$ for the same $n$ and $r$ — every combination of $r$ objects corresponds to exactly $r!$ different permutations, one for each way of ordering that same group.

Deciding which formula a problem needs

The single most important skill in this chapter is recognising, from the wording of a problem, whether order matters:

• Use $^{n}P_{r}$ when the problem asks for arrangements, sequences, rankings, codes, or seatings in a row — situations where swapping two chosen objects gives a genuinely different outcome (e.g. "in how many ways can a 1st, 2nd and 3rd prize be awarded" — different people in different ranks are different outcomes).
• Use $^{n}C_{r}$ when the problem asks for a selection, group, committee, team, or subset — situations where only which objects are chosen matters, not the order in which they were picked (e.g. "in how many ways can 3 students be chosen for a trip" — the same trio chosen in a different order is still the same trio). …