Real-World Combinations Calculator

This page answers three different counting questions, and only the first one is actually a combination. Mixing them up is the most common mistake in this corner of maths, so it is worth being exact about which is which.

Lottery: order does not matter, nothing repeats.

Pick 5 numbers from 69 and it makes no difference whether you wrote 7 first or last, and you cannot pick 7 twice. That is a combination:

C(n, r) = n! ÷ (r! × (n − r)!)

also written nCr, or “n choose r”, where n is the pool size and r is how many you pick.

Powerball draws 5 from 69 white balls, plus 1 red ball from a separate pool of 26: C(69,5) × 26 = 11,238,513 × 26 = 292,201,338, about 1 in 292 million. Set Pool Size to 69, Numbers to Pick to 5, and Bonus Pool to 26 and you will get exactly that.

A handy identity: C(n, r) = C(n, n − r). Choosing 3 from 10 gives the same count as choosing 7 from 10, because naming who is in also names who is out.

PIN: order matters and digits repeat.

1234 and 4321 are different PINs, and 1111 is perfectly legal. So a PIN is not a combination at all. Every position is an independent choice among 10 digits, which makes the count 10^d for d digits. Four digits gives 10,000, six gives a million. Each extra digit multiplies the space by exactly 10. In this mode the pool size is ignored, because a numeric keypad always has ten keys.

Password: the same shape with a bigger alphabet.

For a pool of p characters and a length L the count is p^L. The alphabet matters more than people expect. Eight lowercase letters (p = 26) gives 209 billion. Eight characters drawn from mixed case plus digits (p = 62) gives 218 trillion, over a thousand times more, from the same eight keystrokes.

Combinations against permutations

Permutations P(n,r) = n! ÷ (n−r)! count arrangements where order matters but nothing repeats, which is the case this calculator does not cover. The three cases line up like this: repeats allowed and order matters gives p^L (PIN and password modes), no repeats and order ignored gives C(n,r) (lottery mode), no repeats and order matters gives P(n,r).

Worked example you can run here

Your company has 12 employees and you need a project team of 4. Choose the Lottery mode, set Pool Size to 12, Numbers to Pick to 4, and leave Bonus Pool at 0.

C(12, 4) = (12 × 11 × 10 × 9) ÷ (4 × 3 × 2 × 1) = 11,880 ÷ 24 = 495 possible teams

The brute-force times shown for PINs and passwords assume an offline attack on a stolen hash, at 10,000 guesses per second for PINs and a billion per second for passwords. A live login form with rate limiting is far slower to attack, which is why a 6-digit phone code survives in practice despite the arithmetic here looking alarming.