What is the difference between a permutation and a combination?

A permutation counts ordered arrangements, so ABC and CBA are different. A combination counts unordered selections, so ABC and CBA are the same group. That is why nPr is always greater than or equal to nCr — for any chosen set of r items there are r! ways to order them.

How is nPr calculated?

The permutation formula is nPr = n! / (n − r)!. It is the product of the r largest factors of n!, namely n × (n−1) × … × (n−r+1). For example 10P3 = 10 × 9 × 8 = 720.

How is nCr calculated?

The combination formula is nCr = n! / (r! · (n − r)!), also written as "n choose r" or the binomial coefficient C(n, r). It equals nPr divided by r!. For example 10C3 = 720 / 3! = 720 / 6 = 120.

Why use this calculator instead of a normal one for large numbers?

Factorials grow explosively — 21! already exceeds the largest integer a standard calculator stores exactly, so it silently rounds. This tool uses JavaScript BigInt for every step, so values like 52C5 or 100P10 are exact to the last digit with no floating-point error.

What happens when r is 0 or equals n?

By convention 0! = 1, so nP0 = nC0 = 1 (there is exactly one way to choose nothing) and nCn = 1 (one way to choose everything). nPn = n!, the number of ways to arrange all n items.

Is my data kept private?

Yes. Every calculation runs locally in your browser using built-in BigInt math. No numbers are sent to any server, logged, or stored.

Permutation and Combination Calculator

A permutation and combination calculator that turns two numbers — n (how many items you have) and r (how many you pick) — into both nPr and nCr, then shows the full factorial working behind each answer. It is built for students learning counting principles, teachers checking answers, and anyone working with probability, lottery odds, card hands, or experiment design.

How it works

Counting problems split into two families. When order matters you want a permutation: lining up 3 runners from 10 onto a podium, or generating a 4-digit PIN. The formula is:

nPr = n! ⁄ (n − r)!

When order does not matter you want a combination: dealing a 5-card poker hand, or picking a 6-number lottery ticket. The formula is:

nCr = n! ⁄ (r! · (n − r)!)

The two are linked by a single fact: every unordered group of r items can be arranged in r! different orders, so nPr = nCr × r!. The calculator computes both directly and displays the substituted formula plus each factorial term, so you can follow the arithmetic rather than just trust a number.

Crucially, all maths runs with BigInt. Factorials explode fast — 13! already passes six billion and 21! exceeds the largest integer a normal double can hold exactly. Where ordinary calculators silently round, this tool keeps every digit precise, so 52C5 (the number of distinct poker hands) returns exactly 2,598,960 and far larger inputs stay correct to the last digit.

Worked example

How many ways can a club of 10 members fill the roles of president, secretary and treasurer (order matters), versus simply pick a committee of 3 (order ignored)?

Permutations: 10P3 = 10! / (10 − 3)! = 10! / 7! = 10 × 9 × 8 = 720
Combinations: 10C3 = 10! / (3! · 7!) = 720 / 6 = 120

So there are 720 ordered office assignments but only 120 distinct committees — and indeed 720 = 120 × 3!, confirming the relationship between the two counts.

n	r	nPr (ordered)	nCr (unordered)
5	2	20	10
10	3	720	120
52	5	311,875,200	2,598,960
49	6	10,068,347,520	13,983,816

Formula note

Both formulas rely on the factorial k! = k × (k−1) × … × 2 × 1, with the convention 0! = 1. The combination nCr is also the binomial coefficient that appears in Pascal’s triangle and the expansion of (a + b)ⁿ, where it gives the coefficient of the term aⁿ⁻ʳ bʳ. To avoid building giant intermediate factorials, the tool computes nCr multiplicatively using the smaller of r and n − r, which keeps results exact and fast. Inputs must satisfy 0 ≤ r ≤ n with whole numbers; n is capped at 1000 purely to keep the display responsive.

Every figure is calculated in your browser — no numbers are uploaded or stored.