Count every way to arrange or choose.
Pick your total items and how many you take, then see permutations (order matters), combinations (order does not), factorials, and the with-repetition variants, all at once.
The numbers
The whole set you are choosing or arranging from.
How many you take from the set. Must be r ≤ n.
What these numbers tell you
Four ways to count
Permutations vs combinations, with and without repetition
The count scorecard
Pascal's triangle row for n
Every combination count nCr as r runs from 0 up to n. These are the numbers on row n of Pascal's triangle, and they add up to 2 to the power n.
| r (chosen) | Combinations nCr | Permutations nPr | Share of 2^n |
|---|
Permutations and combinations, explained
Permutations vs Combinations
Permutations count arrangements where order matters (e.g., race placements, passwords). Combinations count selections where order does not matter (e.g., lottery numbers, committee members). The key question is: does rearranging the same items create a different outcome? If yes, use permutations; if no, use combinations.
How to Calculate Permutations (nPr)
The permutation formula is nPr = n! / (n-r)!. For example, arranging 3 items from 10: 10P3 = 10! / 7! = 10 × 9 × 8 = 720. This counts every possible ordered arrangement of r items chosen from n total items.
How to Calculate Combinations (nCr)
The combination formula is nCr = n! / (r! × (n-r)!). For example, choosing 3 items from 10: 10C3 = 10! / (3! × 7!) = 120. This is always less than or equal to the permutation count because it eliminates duplicate arrangements of the same items.
Real-World Applications
Permutations apply to PIN codes, seating arrangements, race outcomes, and any scenario where sequence matters. Combinations apply to lottery odds, card hands, team selection, and survey sampling. Understanding both is essential for probability calculations and counting problems.
Common questions
What is a factorial?
A factorial (n!) is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely fast, 20! is over 2.4 quintillion.
Can r be greater than n?
No. You cannot choose or arrange more items than are available. If r > n, both nPr and nCr are 0 (impossible).
What is the relationship between nPr and nCr?
nPr = nCr × r!. In other words, permutations equal combinations multiplied by the number of ways to arrange r items. This makes sense because permutations count each arrangement separately, while combinations do not.
How do I calculate combinations with repetition?
Combinations with repetition use the formula (n+r-1)! / (r! × (n-1)!). This applies when the same item can be chosen more than once, like scoops of ice cream from multiple flavors.
For counting and probability education. Very large n or r produce numbers beyond exact integer precision; those are shown in scientific notation as close approximations.