Combinations versus permutations
Both count how many ways you can choose r items from n, but they answer different questions. Permutations count arrangements where order matters; combinations count selections where it does not. Picking a 3-digit code is a permutation; picking 3 pizza toppings is a combination.
The formulas
Permutations: nPr = n! divided by (n – r)!. Combinations: nCr = n! divided by r! times (n – r)!. Because combinations ignore the r! ways each selection can be ordered, nCr is always nPr divided by r! — never larger than the permutation count.
Why order changes the count
Choosing A then B differs from B then A as a permutation but is the same combination. That single distinction is why a lock should really be called a permutation lock, and why lottery odds use combinations.
Related tools
To turn these counts into chances, use the probability calculator; for averages of a data set, the mean median mode calculator.
Worked example
From 10 items choosing 3: there are 720 permutations (10 times 9 times 8) but only 120 combinations, since each group of 3 can be ordered in 3! = 6 ways and 720 divided by 6 is 120.
FAQ
What if r is greater than n?
Then there are zero ways — you cannot choose more distinct items than exist — and the calculator returns 0.
Why do huge n and r show scientific notation?
Factorials grow astronomically; beyond about 15 digits the result is shown in exponential form to stay readable.