Why is nCr always smaller than nPr for the same n and r?

Every unordered group of r items corresponds to r! different ordered arrangements. nCr collapses all of those r! permutations into a single count, so nCr = nPr ÷ r!, always making it smaller (or equal, when r ≤ 1).

What does '0! = 1' have to do with this?

By mathematical convention, 0! is defined as 1 (an empty product), which makes the formulas work correctly at the edges — e.g. choosing all n items from n (r = n) or choosing zero items (r = 0) both correctly give exactly 1 way.

Why does this calculator show 'approximately' for very large n?

Once the true result exceeds Number.MAX_SAFE_INTEGER (2^53 − 1), JavaScript's standard number type can no longer represent every integer in that range exactly — so the result is shown in scientific notation as an explicit approximation, rather than printing digits that would look precise but might not be.

Can r be larger than n?

No — you can't arrange or choose more items than exist, so the calculator rejects r > n as invalid input.

Can n or r be negative or a fraction?

No — both must be non-negative whole numbers, since 'choosing 2.5 items' or 'choosing -1 items' has no combinatorial meaning.

Where are permutations and combinations used in real life?

Probability calculations (like lottery odds, which use nCr), password/PIN entropy estimates (which use nPr when repetition isn't allowed), scheduling and seating arrangements, and statistical sampling.

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Math & Everyday

Permutations & Combinations Calculator

nPr and nCr for any n and r, computed without overflowing on large factorials.

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About the Permutations & Combinations Calculator

Computes nPr (permutations — order matters) and nCr (combinations — order doesn't matter) for any whole numbers n and r, using an iterative multiply-and-divide method that stays numerically stable well beyond where a literal factorial calculation would overflow.

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How to use it

Enter n (the total number of items) and r (how many you're choosing or arranging).
Read both nPr (permutations) and nCr (combinations) in the result panel.

Formula

Permutations: nPr = n! ÷ (n − r)!
Combinations: nCr = n! ÷ (r! × (n − r)!)

Computed here as an iterative product (multiplying and dividing term-by-term) rather than by forming n! directly — this avoids both the precision drift that starts creeping into a literal factorial around 19! and the overflow to Infinity that hits standard double-precision numbers past 171!.

Worked example

Choosing and ranking a 1st, 2nd, and 3rd place from 10 runners (order matters): nPr = 10 × 9 × 8 = 720 permutations. Choosing an unordered group of 3 runners to advance from the same 10 (order doesn't matter): nCr = 720 ÷ 3! = 120 combinations.

Permutations vs. combinations — when to use each

Permutations vs. combinations — when to use each
Scenario	Order matters?	Use
Ranking 1st/2nd/3rd place	Yes	nPr
Choosing a committee of 3 from 10 people	No	nCr
Assigning distinct prizes to winners	Yes	nPr
Picking lottery numbers (as a set)	No	nCr
Arranging books on a shelf	Yes	nPr

Interpreting your result

n and r must both be non-negative whole numbers, with r ≤ n (you can't choose or arrange more items than exist). For very large n, the true result can exceed JavaScript's safe integer range (2^53 − 1, about 9 quadrillion) — past that point, the exact digit-for-digit value can't be represented precisely by a standard number, so the result is shown in scientific notation as an explicit approximation rather than a string of digits that would look exact but isn't.

Recommendations

• The key question to ask before picking a formula: does swapping the order of two chosen items create a genuinely different outcome? If yes, use nPr; if no, use nCr.
• nCr is always less than or equal to nPr for the same n and r, since every combination corresponds to r! different permutations (all the ways to order that same group).
• nCr is symmetric: choosing r items from n is the same count as choosing the (n − r) items you'd leave out — nCr(n, r) = nCr(n, n − r).

Frequently asked questions

A permutation counts arrangements where order matters (ABC is different from CBA); a combination counts selections where order doesn't matter (ABC and CBA count as the same group).