Order is the dividing line
Both formulas count ways to pick r things from n, but they answer different questions. A permutation treats AB and BA as two distinct outcomes because the order matters — think of finishing positions in a race or arranging letters into a code. A combination treats AB and BA as the same choice because only the membership of the group matters — think of dealing a hand of cards or choosing a committee.
That is why nPr is always at least as large as nCr: every combination corresponds to several permutations, one for each way its members can be ordered.
From one formula to the other
The permutation count is n! divided by (n−r)!, which cancels the arrangements of the items you did not pick. To get combinations, divide that result by r! to collapse all the orderings of the chosen r items into a single selection.
- nPr = n! / (n−r)! counts ordered arrangements.
- nCr = nPr / r! removes the duplicate orderings.
- The chart compares the two counts so you can see how much the order requirement inflates the total.
Practical notes and pitfalls
Use permutations when sequence carries meaning — passwords, rankings, seating in a row, scheduling. Use combinations when it does not — lottery numbers, pizza toppings, sampling without regard to order. A frequent mistake is reaching for permutations when the question is really about an unordered group, which overcounts by a factor of r!.
These counts assume each item is used at most once. If items may repeat (such as digits in a PIN), use the combinatorics-with-repetition tool instead. Factorials also grow extremely fast, so very large n and r can exceed the range of exact whole-number arithmetic.
Formula
nPr = n! / (n−r)!; nCr = n! / (r!·(n−r)!)Frequently asked questions
- What is the difference between a permutation and a combination?
- A permutation counts arrangements where order matters; a combination counts selections where order does not. For the same n and r, there are always at least as many permutations as combinations.