Permutations or combinations?

Permutations when order matters; combinations when it doesn’t.

permutations() — Permutations

Counts the ways to put things in order.

nPr = n! / (n − r)!

A permutation P(n,r) counts ways to arrange r of n items when ORDER MATTERS. (e.g. P(5,2) = 20)
Formula: P(n,r) = n! / (n-r)!, where n! means n times every whole number below it. (e.g. P(4,2) = 24/2 = 12)
Permutations are always at least as many as combinations (where order does not matter). (e.g. P(5,2)=20 >= C(5,2)=10)
Arranging all n items is just n! (the case where r = n). (e.g. P(4,4) = 4! = 24)
Use it for ordered outcomes like 1st and 2nd place finishers. (e.g. 5 runners, top 2 -> P(5,2) = 20)
Circular permutation: around a round table only the order RELATIVE to each other matters, since spinning everyone gives the same seating, so n people arrange in (n-1)! ways. (e.g. 4 around a table -> (4-1)! = 6)
If direction is NOT distinguished either — a seating and its mirror-image flip count as the same, like beads on a necklace — halve it once more: (n-1)!/2. (e.g. 4 beads on a necklace -> 3!/2 = 3)

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

$P (5, 3) = \frac{5 !}{( 5 - 3 )!} = \frac{120}{2} = 60$

Think of a race: who gets gold, silver, and bronze? Swapping two runners makes a new result, so the order counts.

In a race of 5 runners, you hand out gold, silver, and bronze. Since order matters, the number of ways is permutations(5, 3) = 5 × 4 × 3 = 60 different podiums.
A club of 10 members must choose a president and a vice-president. Because the two roles are different, there are permutations(10, 2) = 90 possible pairings.
A 4-digit PIN using digits 0–9 with no repeats has permutations(10, 4) = 5040 possibilities, since each position must be a different digit and order counts.
Seating 5 friends around a round table is a circular permutation: rotating everyone is the same seating, so there are only (5−1)! = 24 arrangements — far fewer than the 120 you would get in a straight row.

With 5 runners, the number of ways to hand out gold, silver, and bronze (pick 3 in order) is 60.

Permutations or combinations?: Permutations when order matters; combinations when it doesn’t.