Permutations and Combinations Calculator

How to calculate Permutations and Combinations?

Permutations (nPr) are used to count the number of ways to arrange items where the specific order is important, such as a race finish or a seat assignment. Combinations (nCr) are used to count the number of ways to pick a group from a larger set where the order of selection does not change the outcome, such as picking a committee or a hand of cards.

The formula for permutations is nPr = n! / (n - r)! and the formula for combinations is nCr = n! / [r! * (n - r)!].

Using the Combinatorics Calculator: an example

Suppose you have a set of 5 total items (n=5) and you want to select 3 items (r=3) from that set.

Step-by-step calculation:

1. Calculate the factorials for n, r, and (n-r): 5! = 120, 3! = 6, and 2! = 2.
2. Divide n! by (n-r)! to get the Permutations (nPr): 120 / 2 = 60.
3. Further divide the Permutations result by r! to get the Combinations (nCr): 60 / 6 = 10.

What if n is less than r?

In standard combinatorics, it is impossible to choose more items than are available in the set; therefore, the result for both nPr and nCr will be 0.

Why is 0! equal to 1?

By mathematical definition, 0! equals 1 to ensure that the formulas for permutations and combinations work correctly even when selecting zero or all items from a set.

When should I use combinations instead of permutations?

Use combinations when the result of the selection is the same regardless of order, such as choosing three flavors of ice cream for a single bowl.