Permutation vs Combination: Which One You Actually Need

Almost every mistake with these two formulas starts in the same place. People reach for the calculator before deciding whether order counts. Settle that first and the arithmetic is trivial.

The question that picks the formula for you

Ask whether swapping two of your chosen items changes the outcome. If Alice-then-Bob is not the same result as Bob-then-Alice, order matters and you want a permutation. If they end up on the same committee either way, order does not matter and you want a combination.

Seating people in a row, ranking finishers, setting a passcode, dealing cards into named positions: permutations. Choosing pizza toppings, picking lottery numbers, forming a team, drawing a poker hand you will fan out anyway: combinations.

How the two are related

Combinations are permutations with the duplicate orderings divided out. Any group of 3 can be arranged 3! = 6 ways, so the permutation count is exactly six times the combination count. With the 10 runners, 720 divided by 6 is 120. That factor of r factorial is why the two answers drift so far apart as r grows.

Two worked examples

A lottery draw

Six balls from 49, order irrelevant. The ordered count is 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520. Divide that by 6! = 720 and you get 13,983,816 combinations. That second figure is how many tickets you would have to buy to cover every possible result, and it is the honest way to describe the odds.

A four-digit PIN

If digits can repeat, every position has 10 choices: 10 x 10 x 10 x 10 = 10,000 codes. If you insist on four different digits it becomes 10 x 9 x 8 x 7 = 5,040. Banning repeats throws away half the possible codes, which surprises people who assume unique digits are the safer choice.

Using the permutation and combination calculator

Two inputs do the work. The field marked n is the size of the pool you are drawing from, and r is how many you take. Then pick the mode: nPr for arrangements, nCr for selections. If the same item can be chosen more than once (a PIN, a dice roll, a drinks menu with refills) switch on the repetition option, which changes the maths to n to the power r for ordered picks.

Answers come back as exact whole numbers rather than rounded ones, which matters because these counts grow fast. Choosing 10 from 20 with order gives over 670 billion. If your result appears in scientific notation you have probably entered n and r the wrong way round. Once you have a count, a probability calculator turns it into odds, and if you want to simulate draws rather than count them, a random number generator is the right tool instead.

Common questions

Is a combination lock really a combination? No. 12-34-8 will not open a lock set to 8-12-34, so order matters and it is a permutation. The name is historical sloppiness. Worth knowing if you are estimating how long a lock takes to work through by hand.

What do nPr and nCr mean on my calculator? P is permutation, C is combination, n is the pool, r is the pick. On most models you type n, press the function key, then type r. Some label them P(n,r) and C(n,r) instead, which is clearer.

Can r be bigger than n? Not without repetition. You cannot choose 5 people from 3. With repetition allowed it is fine, and a 6-digit code drawn from 10 digits has r = 6 against n = 10.

How many five-card poker hands are there? Five cards from 52, and once they are in your hand the order is irrelevant, so it is a combination: 2,598,960. The ordered count is 52 x 51 x 50 x 49 x 48 = 311,875,200, and dividing that by 5! = 120 brings you back to the combination figure.

Why does 0 factorial equal 1? Because there is exactly one way to arrange nothing. It is a definition chosen so the formulas keep behaving at the edges, and it stops nCn from dividing by zero.