Permutation & Combination Calculator

Calculate nPr permutations and nCr combinations

Written by toolforge.websiteLast reviewed 23 August 2026How we build and check these tools

n (total items)

r (items to choose)

Results

Permutation (nPr)

—

Combination (nCr)

—

Permutation Formula:

nPr = n! / (n-r)!

10P3 = 10! / 7! = —

Combination Formula:

nCr = n! / (r! × (n-r)!)

10C3 = 10! / (3! × 7!) = —

Factorial Breakdown:

10! = —

(7)! = —

Step-by-step:

Permutation:

1.n! = 10! = —
2.(n-r)! = 7! = —
3.nPr = — / — = —

Combination:

1.n! = 10! = —
2.r! = 3! = —
3.(n-r)! = 7! = —
4.nCr = — / (— × —) = —

Permutation & Combination Calculator: key facts

What it does: Calculate nPr permutations and nCr combinations
Category: Math Calculators
Cost: Free, with no account, sign-up, or install.
Your data: Runs entirely in your browser — the files and text you enter are never uploaded to a server.
Last reviewed: 23 August 2026. Report an incorrect result.

Understanding the Permutation & Combination Calculator

When you need to count arrangements or selections, the difference between a permutation and a combination is everything — and it is easy to mix them up. This calculator computes both at once from the same two inputs: n, the total number of items, and r, how many you are choosing.

Permutations (nPr) count ordered arrangements — useful for problems like how many ways a race of n runners can finish in the top r places. Combinations (nCr) count unordered selections — how many r-person committees you can form from n people, or how many lottery tickets cover a draw. Students checking probability homework and anyone setting up a counting problem get both answers plus the factorial breakdown behind them.

Getting a result

Enter n, the total number of items, in the first field.
Enter r, the number of items being chosen, in the second field.
Read the permutation (nPr) and combination (nCr) results, both calculated automatically.
Expand the factorial breakdown to see n!, r!, and (n−r)! and how they combine.
Use Copy to save both formulas and values, or Reset to clear and start over.

Whether order matters

Both counts start from factorials and differ by one division. A permutation counts ordered arrangements: n factorial divided by the factorial of n minus r. A combination counts unordered selections, dividing additionally by r factorial to collapse every ordering of the same chosen set into one.

That extra divisor is the entire distinction, and it is always the question to settle first. A podium finish is a permutation, because gold and silver are different outcomes. A hand of cards is a combination, because the order you were dealt them makes no difference. Permutations therefore always outnumber combinations for the same n and r, by exactly r factorial.

The tool shows the intermediate factorials so the arithmetic is auditable rather than a single opaque number.

P(n, r) = n! ÷ (n − r)! C(n, r) = n! ÷ (r! × (n − r)!)

The defaults — n = 10, r = 3 — give 720 permutations and 120 combinations.
The ratio between them is exactly 3! = 6, the number of ways to order any three chosen items.
Picking 6 numbers from 49 is a combination: 13,983,816 possibilities, which is where lottery odds come from.

Reasons to use it here

It computes both nPr and nCr simultaneously, so you can compare ordered and unordered counts side by side instead of running two tools.
Each answer is shown with its formula and a factorial breakdown — n!, r!, and (n−r)! — making it a genuine learning aid, not just a black-box number.
It validates the inputs, returning zero when r exceeds n or when a value is negative, so impossible cases do not produce misleading output.
Large results are formatted with thousands separators and computed instantly in the browser, with no sign-up.

Repetition, and how fast factorials grow

Both formulas here assume selection without replacement — each item can be chosen once — and that every item is distinguishable. Different formulas apply when repetition is allowed, and arrangements of a set containing identical items require dividing by the factorial of each repeated group, which is why the letters of a word with doubled letters have fewer distinct arrangements than its length would suggest.

Factorial growth is faster than most people expect and it has a practical consequence here. 20 factorial already exceeds two quintillion, and beyond about 170 factorial the value overflows the range of a double-precision number entirely. Because the tool computes the factorials directly rather than cancelling terms first, large inputs lose precision before they lose validity — so treat very large results as approximate.

One elegant property worth knowing: choosing r from n is always the same count as choosing n − r from n, because selecting a group is equivalent to selecting everyone left out. Choosing 47 from 49 is as easy to compute as choosing 2.

Frequently Asked Questions

What is the difference between a permutation and a combination?

A permutation counts arrangements where order matters (nPr); a combination counts selections where order does not (nCr). Choosing a president and vice-president from a group is a permutation; choosing two committee members is a combination. For the same n and r, there are always at least as many permutations as combinations.

What does the factorial notation n! mean?

n! ("n factorial") is the product of all positive integers up to n — for example 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! and 1! both equal 1. Factorials are the building blocks of the permutation and combination formulas shown in the breakdown.

Why do I get 0 as a result?

The calculator returns 0 when the inputs describe an impossible selection — most commonly when r is larger than n, or when either value is negative. You cannot choose more items than exist, so check that r is no greater than n.

What are the formulas used?

Permutations use nPr = n! ÷ (n−r)!, and combinations use nCr = n! ÷ [r! × (n−r)!]. The tool displays both formulas with your numbers substituted in so you can verify each step.