How do I remember which formula has the extra r! in the denominator?

Combinations always have the extra r! because they collapse the r! duplicate orderings of each group down into a single count. Permutations, which count each ordering separately, do not divide by r! — so C(n,r) is always P(n,r) divided by r!, and C(n,r) ≤ P(n,r) for any r > 1.

What does 5! (factorial) actually mean?

5! (read "5 factorial") means 5 × 4 × 3 × 2 × 1 = 120 — the number of ways to arrange 5 distinct items in a row. Factorials appear in both the permutation and combination formulas because arranging or selecting from a set fundamentally involves counting sequences of shrinking choices (5 options, then 4 remaining, then 3, and so on).

Are permutations always a larger number than combinations for the same n and r?

Yes, whenever r ≥ 2 — permutations are always exactly r! times larger than combinations for the same n and r, since permutations count every internal ordering of each group separately while combinations count each group exactly once. When r = 0 or r = 1, they are equal, since there is only one way to order 0 or 1 item.

Is choosing a committee a permutation or a combination problem?

A combination, as long as every committee member has the same role. If the "committee" actually assigns distinct roles (chair, secretary, treasurer) to different people, then the order/role assignment matters and it becomes a permutation problem instead — this is a common source of exam mistakes.

Does the formula change if repetition is allowed (choosing the same item more than once)?

Yes — both formulas above assume selection without repetition. With repetition allowed, permutations with repetition use n^r (n choices independently made r times), and combinations with repetition use the "stars and bars" formula C(n+r−1, r). Always confirm whether a problem allows repeated selections before applying the standard formulas.

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Permutations vs. Combinations

The entire difference comes down to one question: does the order of selection change the outcome?

Both permutations and combinations count the number of ways to choose r items from a set of n — the difference is entirely about whether order matters. A permutation counts arrangements where order matters: picking A-then-B is a different outcome from B-then-A. A combination counts selections where order doesn't matter: {A, B} is the same group regardless of which one was picked first. Because every ordered arrangement can be unwound into the same unordered group in r! ways, permutations always outnumber combinations for the same n and r, by exactly a factor of r!.