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Permutation Combination Calculator

Permutation Combination Calculator is part of Toolspire and is ready as soon as this page loads. nPr is the arrangements of r items from n. nCr is the groups, where order inside the group does not matter. Both use factorial division that stays exact.

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What you can do here

Both counts on one page

The headline is nPr. The detail is nCr. You can see that combinations are the permutations divided by r!.

Exact division

BigInt factorials divide evenly for these formulas, so 5C2 is 10 and not 9.999.

Refuses r greater than n

Choosing 6 items from 5 is not given a zero disguised as a result. It is rejected.

The sequence

How to use the Permutation Combination Calculator

  1. 1

    Decide whether order matters

    A podium (gold, silver, bronze) is a permutation. A committee of three, with no roles, is a combination.

  2. 2

    Type n first and r second

    n is the size of the pool. r is how many you pick. Swapping them usually makes the page reject the input.

  3. 3

    Check 5 and 2

    You should see nPr 20 and nCr 10 before you use the page on a larger class list.

Worth knowing

  • Permutation Combination Calculator on Toolspire: nPr = n! / (n − r)!. It counts arrangements, so order matters.
  • nCr = nPr / r!. It counts groups, so {A, B} and {B, A} are one combination.
  • 5P2 = 20 and 5C2 = 10. Those are the checks built into the test.
  • n and r must be integers, r cannot exceed n, and n stops at 500 so the factorials stay bounded.

When to divide by r!

Each combination of r items can be arranged in r! orders. Counting arrangements first (nPr) and then dividing by r! removes those orders and leaves the combination. If the question says “arrange”, “rank” or “password”, keep nPr. If it says “choose”, “committee” or “hand of cards”, use nCr. The page shows both so the extra division is visible.

FAQ

Questions about Permutation Combination Calculator

Where does Permutation Combination Calculator run?

Permutation Combination Calculator runs in this tab on Toolspire. What you type or drop in is not sent to us and is not stored on a server.

What is the difference between nPr and nCr?

nPr counts ordered selections. nCr counts unordered ones. nCr is nPr divided by r factorial.

What is 5C2?

10. The pairs from five items are ten, because order inside a pair is not a new pair.

Why is nCn equal to 1?

There is one way to choose every item. nPn is n!, the arrangements of all of them. The page shows both.

Can r be zero?

Yes. There is one way to choose nothing, and one way to arrange nothing, so both results are 1.

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