Statistics & Probability

Permutations vs Combinations: When Order Matters

By Math Solving Space · · 2 min read

On this page
  1. The key question: does order matter?
  2. Factorials
  3. Permutations
  4. Combinations
  5. Worked example: the lottery
  6. Combinations in probability
  7. Repetition allowed
  8. Common mistakes
  9. Practice questions
  10. Frequently asked questions

A permutation counts arrangements where order matters, like first, second and third place. A combination counts selections where order does not matter, like choosing a team. Choosing 3 from 10 gives 720720 permutations but only 120120 combinations, because each group of 3 can be arranged in 3!=63! = 6 ways.

The key question: does order matter?

Situation Order matters? Use
Gold, silver, bronze from 10 runners Yes Permutation
A committee of 3 from 10 people No Combination
A 4-digit PIN Yes (and digits can repeat) 10410^4
Lottery numbers No Combination

Factorials

n!n! means n×(n−1)×⋯×1n \times (n-1) \times \cdots \times 1. For example, 5!=1205! = 120, and by definition 0!=10! = 1.

Permutations

nPr=n!(n−r)!{}^{n}P_{r} = \frac{n!}{(n - r)!}

Example: medals for 10 runners.

10P3=10×9×8=720{}^{10}P_{3} = 10 \times 9 \times 8 = 720

There are 10 choices for gold, then 9 left for silver, then 8 for bronze.

Combinations

(nr)=n!r! (n−r)!\binom{n}{r} = \frac{n!}{r!\,(n - r)!}

Example: a committee of 3 from 10.

(103)=7203!=7206=120\binom{10}{3} = \frac{720}{3!} = \frac{720}{6} = 120

Worked example: the lottery

Choosing 6 numbers from 49, where order does not matter:

(496)=13 983 816\binom{49}{6} = 13\,983\,816

So one ticket has a 1 in about 14 million chance of matching all six.

Combinations in probability

Combinations appear in the binomial formula. The probability of exactly 6 heads in 10 fair coin flips is

(106)(12)10=2101024≈0.2051\binom{10}{6}\left(\tfrac{1}{2}\right)^{10} = \frac{210}{1024} \approx 0.2051

Repetition allowed

  • Order matters, repetition allowed: nrn^r (a 4-digit PIN has 104=10 00010^4 = 10\,000 options).
  • Order doesn't matter, repetition allowed: (n+r−1r)\binom{n + r - 1}{r}.

Common mistakes

  • Using permutations for a team or a hand of cards.
  • Forgetting that (nr)=(nn−r)\binom{n}{r} = \binom{n}{n - r}, which can save work.
  • Writing n!r!\frac{n!}{r!} and forgetting the (n−r)!(n - r)!.

Practice questions

  1. How many ways can 5 books be arranged on a shelf?
  2. How many 5-card hands come from a 52-card deck?
  3. How many ways to pick a president and a secretary from 8 people?

Answers: 1) 120120 2) 2 598 9602\,598\,960 3) 5656

The permutation and combination calculator gives exact answers even for huge numbers. For "exactly k successes" questions, use the binomial distribution calculator.

Frequently asked questions

Why is 0! equal to 1?

There is exactly one way to arrange nothing, and defining 0! = 1 makes the formulas work for r = 0 and r = n.

Is nCr always smaller than nPr?

It is never bigger. They are equal only when r is 0 or 1.

How do I know which formula to use?

Ask whether swapping two chosen items gives a different outcome. If yes, use permutations; if no, use combinations.

Open the calculator →

#probability#counting#combinations

Related guides