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Permutations & Combinations

🎲 Probability & Statistics
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Permutations & Combinations

Count ordered arrangements and unordered selections using P(n,r) and C(n,r).

A permutation is an ordered arrangement of objects. A combination is an unordered selection. The key question: does order matter? If yes, use permutations. If no, use combinations.

P(n,r)=n!(n−r)!C(n,r)=(nr)=n!r! (n−r)!P(n, r) = \frac{n!}{(n-r)!} \qquad C(n, r) = \binom{n}{r} = \frac{n!}{r!\,(n-r)!}
n!=n⋅(n−1)⋅(n−2)⋯2⋅10!=1n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1 \qquad 0! = 1

Permutations (order matters)

Number of ways to arrange r from nP(5,3) = 5!/(5−3)! = 60Lock combinations, race finishesLonger formula (no r! in denominator)

Combinations (order irrelevant)

Number of ways to choose r from nC(5,3) = 5!/(3!·2!) = 10Poker hands, committee selectionShorter count: C(n,r) = P(n,r)/r!

C(n,r) = P(n,r)/r! — dividing by r! removes the overcounting of the r! arrangements of the chosen items.

🏆Choosing a Committee
From 8 people, how many 3-person committees? Order doesn't matter: C(8,3) = 8!/(3!·5!) = (8·7·6)/(3·2·1) = 56.
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Remember This!

Pascal's Triangle gives binomial coefficients: C(n,r). Note C(n,r) = C(n, n−r) — choosing r to include is the same as choosing n−r to exclude.

✏️ Try It!

How many ways can 4 books be arranged on a shelf from a collection of 7?

Practice with CalcVerse
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