The formula
How it works
Count the number of ways to choose or arrange items. Permutations count arrangements where order matters; combinations count selections where it does not. Enter how many items you have and how many you are picking.
FAQ
When do I use permutations vs combinations?
Use permutations when order matters — a race podium, a PIN, a ranked list. Use combinations when it does not — a lottery draw, a hand of cards, a team of people picked from a group.
Why are there always fewer combinations?
Because each combination corresponds to several permutations — one for every way of ordering the same chosen items. Dividing the permutations by r! removes those duplicate orderings.
What happens when r equals n?
The number of combinations is always 1, since there is only one way to choose every item, while the number of permutations becomes n! — every possible ordering of the full set.
What happens when r is 0?
Both nPr and nCr equal 1 — there is exactly one way to choose or arrange nothing at all, an empty selection.
Why do the results grow so quickly as n increases?
Factorials grow explosively — 10! is already over 3.6 million — so even modest increases in n or r can push the counts into the millions or billions.
What does 0! equal, and why does it matter here?
0! is defined as 1, which is what keeps the formulas working correctly at the edges, such as when r equals n and (n−r)! becomes 0!.
Can n or r be negative or non-integer?
No — both formulas assume n and r are non-negative whole numbers, since you cannot choose a fractional or negative number of items.
About the permutation and combination calculator
This calculator counts how many ways you can choose or arrange a number of items — the two most important counting tools in combinatorics. Permutations count arrangements where the order matters, and combinations count selections where it does not. Both come up constantly in probability, statistics and everyday “how many ways” questions, and the numbers grow surprisingly fast, which is exactly why a calculator helps.
How to use it
Enter the total number of items you are choosing from, n, and how many you are picking, r. The calculator returns both the number of permutations and the number of combinations. For example, choosing 2 from 5 gives 20 permutations but only 10 combinations, because each pair can be ordered two ways. Make sure r is no larger than n — you cannot pick more items than you have.
The formula
Permutations are and combinations are , where the exclamation mark is the factorial (the product of all whole numbers up to that value). The only difference is the extra in the denominator for combinations, which cancels out the different orderings of each chosen group — since order does not matter, they should only be counted once.
Where it is used
These counts are the backbone of probability. Lotteries and card games use combinations to work out the odds, since order does not matter; passwords, race results and seating plans use permutations, since it does. Statisticians use them to build probability distributions, and they appear in scheduling, cryptography and any problem that asks how many possible outcomes there are. Getting the counting right is the first step in almost every probability calculation.