permvscomb() — Permutation vs Combination

The one question that decides which to use: does order matter?

P(n, r) vs C(n, r)

Key idea

• Ask one question first: does ORDER matter? Yes → permutation. No → combination. (e.g. podium vs team)
• Permutation P(n,r) counts ordered arrangements; combination C(n,r) counts unordered groups. (e.g. P(5,3)=60, C(5,3)=10)
• They are linked by P(n,r) = C(n,r) × r!, because each group can be reordered r! ways. (e.g. 60 = 10 × 3!)
• So a permutation count is always ≥ the matching combination count. (e.g. P(n,r) ≥ C(n,r))
• Permutations: passwords, rankings, seatings. Combinations: teams, committees, card hands, lotteries. (e.g. PIN vs lottery)

Definition

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

$n$

total items to choose from

$r$

how many you take

$P(n,r)$

ordered arrangements (order matters)

$C(n,r)$

unordered selections (order ignored)

$$P(5,3)=60,C(5,3)=10,P=C\times r!=10\times 3!=10\times 6=60$$

In plain words

Order matters? Use a permutation (P). Order doesn’t? Use a combination (C). They differ by exactly r! — the number of ways to shuffle each pick.

Where you'd use it

• A 4-digit PIN from 0–9 (no repeats) cares about order, so it is a permutation: P(10,4) = 5,040.
• Picking a 3-person team from 10 does not care about order, so it is a combination: C(10,3) = 120.
• Same pick, both views: choosing 3 of 5 gives C(5,3) = 10 groups, but P(5,3) = 60 ordered line-ups — exactly 3! = 6 times more.

Frequently asked questions

How do I decide which one to use?

Ask one question: if I shuffle my picks, is it a new outcome? Yes → permutation (order matters). No → combination (order ignored).

Why are there always more permutations?

Every group of r items can be arranged in r! orders, so nPr = nCr × r!. Permutations count each of those orders separately.

Is a "combination lock" a combination?

Ironically, no — 1-2-3 and 3-2-1 open different locks, so order matters. Mathematically it is a permutation lock.

Which one is a lottery ticket?

A combination: the 6 numbers you pick win no matter what order the balls are drawn in.