Permutation and combination calculator

Count the ways to choose or arrange r items from n, with or without repetition, and see exactly why with every step and an optional list of each arrangement.

  • Free, no sign-up
  • Step-by-step working
  • Runs in your browser

Not sure this is the right tool? Find the right one

  1. 1 Enter your data
  2. 2 Choose an option
  3. 3 Click “Calculate”
  4. 4 Read the result

Tip: You can paste values copied from Excel, Google Sheets or a report instead of typing them.

These are example values so you can see how the tool works. Replace them with your own, then press the button.

How many items, and how many to pick?

Does order matter? Can items repeat?

Type of count

List the arrangements (optional)

For example A, B, C, D, E. Leave blank to just count.

Your answer

How to use the permutation and combination calculator

  1. Enter the total number of items (n) and how many are chosen (r).
  2. Pick the type: does order matter, and can items repeat? Each option shows its notation.
  3. Optionally type your items, like names or letters, to see every arrangement listed.
  4. Click Calculate. The answer comes with all four counts side by side, so you can see how the choice changes the result.
  5. Follow the slot diagram and steps to see where the formula comes from.

Permutation and combination formulas

nPr = n!(n − r)! , nCr = n!r! (n − r)!

With repetition allowed:

permutations = nr , combinations = (n + r − 1)!r! (n − 1)!

n
The total number of items to choose from
r
The number of items chosen or arranged
n!
“n factorial”: n × (n − 1) × … × 2 × 1, with 0! = 1
nPr
Permutations: ordered arrangements, no repeats
nCr
Combinations: unordered selections, no repeats. Also written C(n, r) or “n choose r”

Worked example

A club of 5 members picks a president, vice president and secretary. Then, separately, it picks a 3-person committee. How many ways can each be done?

  1. Officers: order matters (president ≠ secretary), so it's a permutation. 5 choices, then 4, then 3
    5P3 = 5 × 4 × 3 = 60
  2. Committee: order doesn't matter. Each group of 3 was counted 3! = 6 times above, once for each order
    5C3 = 603! = 606 = 10

What it means: there are 60 ways to fill the three officer roles but only 10 different committees, because the committee {Anna, Ben, Cara} is the same however you order the names. Dividing by r! to remove the order is the whole difference between permutations and combinations.

How to decide which one you need

  • Rankings, positions, codes, passwords, seating: order matters, so use permutations.
  • Teams, committees, hands of cards, lottery numbers, pizza toppings: order doesn't matter, so use combinations.
  • Can the same item be used again? Digits in a PIN and letters in a password can repeat; people on a committee can't.
  • Probability: divide the number of favorable outcomes by the total. For example, the chance of winning a lottery that draws 6 numbers from 49 is 1 / ₄₉C₆ = 1 in 13,983,816.

Counting is the first step in many probability problems. For the probability of a number of successes in repeated trials, use the binomial distribution calculator, which uses nCr in its formula.

Frequently asked questions

What is the difference between a permutation and a combination?

In a permutation the order matters: first, second and third place are different outcomes, so ABC and CBA count separately. In a combination the order doesn’t matter: choosing toppings A, B and C is the same pizza in any order. A quick test: if swapping two chosen items gives a different result, use permutations; if not, use combinations.

What do nPr and nCr mean?

nPr is the number of permutations of r items chosen from n: n! / (n − r)!. nCr (also written C(n, r) or “n choose r”) is the number of combinations: n! / (r!(n − r)!). For example, 5P3 = 60 ordered ways to fill 3 places from 5 people, and 5C3 = 10 ways to choose a group of 3.

When do I use “with repetition”?

When the same item can be chosen more than once. A 4-digit PIN uses digits with repetition and order matters, so it is a permutation with repetition: 10⁴ = 10,000. Choosing 3 scoops from 5 ice cream flavors where you can repeat a flavor, and the order in the bowl doesn’t matter, is a combination with repetition: C(5 + 3 − 1, 3) = 35.

What is 0! (zero factorial)?

0! = 1. It makes the formulas work at the edges: there is exactly one way to choose nothing (nC0 = 1) and one way to choose everything (nCn = 1). It also matches the pattern n! = n × (n − 1)!, since 1! = 1 × 0!.

How do I calculate nCr and nPr on a TI-84?

Type n, then press MATH, move right to PROB, and choose 2:nPr or 3:nCr, then type r and press ENTER. For 5C3, that’s 5 nCr 3 = 10. Factorials are 4:! on the same menu.

How do I calculate combinations in Excel or Google Sheets?

Use =COMBIN(n, r) for combinations and =PERMUT(n, r) for permutations. With repetition, use =COMBINA(n, r) and =PERMUTATIONA(n, r) (or simply =n^r). =FACT(n) gives n!. Excel switches to scientific notation for very large results; this calculator gives the exact number.

Last reviewed November 13, 2026