Order is the only difference
A permutation counts arrangements: choosing a president, secretary and treasurer from ten people gives 720 outcomes, because who holds which role matters. A combination counts selections: choosing three people for a committee gives 120, because the committee is the same regardless of the order picked.
The two are related by a factor of r!. Every combination of r items can be arranged in r! different orders, which is exactly the ratio shown above.
With and without repetition
The formulas here assume each item can be chosen only once — drawing names from a hat without replacement. If repetition is allowed, as with a PIN where digits can repeat, the count is simply n raised to the power r: a four-digit PIN gives 10⁴ = 10,000 possibilities.
Deciding which case applies is usually where the mistake happens. Ask whether an item, once used, is still available for the next selection.
Combinations in probability
Lottery odds are a combination problem. Choosing 6 numbers from 49 without regard to order gives 49C6 = 13,983,816 possible tickets, so a single ticket has roughly a one in fourteen million chance.
Card probabilities work identically. There are 52C5 = 2,598,960 distinct five-card poker hands, which is the denominator behind every hand probability in the game.
Frequently asked questions
Why is 0! equal to 1? +
There is exactly one way to arrange nothing — the empty arrangement. Defining 0! as 1 also keeps the nCr formula working when r equals 0 or n.
Is a combination lock a combination? +
No — it is a permutation, since the order of the digits matters. The name is a long-standing misnomer.
Why does nCr get smaller than nPr? +
Because combinations collapse all r! orderings of the same selection into one outcome, so the count is smaller by exactly that factor.