Combination and Permutation Calculator

What This Calculator Does

Both answer “how many ways can I pick $r$ things from $n$?” - but they split on whether the order matters:

• A permutation counts arrangements where order matters. Awarding gold, silver, and bronze to 8 runners: swapping who gets gold and silver is a different result.
• A combination counts groups where order does not matter. Picking 3 friends for a team: it is the same team no matter what order you name them.

Because order creates extra possibilities, there are always at least as many permutations as combinations. Both are built out of factorials.

The Formulas

$$C(n, r) = \frac{n!}{r!\,(n - r)!} \qquad P(n, r) = \frac{n!}{(n - r)!}$$

A permutation counts arrangements (order matters); a combination counts selections (order does not). So P(n, r) is always at least as large as C(n, r).

How to Use It

1. Choose combinations or permutations.
2. Enter n (total items) and r (how many are chosen).
3. Read the result and the steps.

Worked Examples

n	r	Combinations C(n, r)	Permutations P(n, r)
5	2	10	20
6	2	15	30
10	3	120	720
52	5	2,598,960	311,875,200

FAQ

What is the difference between a combination and a permutation?

A combination counts selections where order does not matter; a permutation counts arrangements where it does. Choosing 2 letters from A, B, and C gives 3 combinations but 6 permutations.

How do you calculate nCr?

Use $C(n, r) = \frac{n!}{r! \times (n - r)!}$. For C(5, 2): $\frac{5!}{2! \times 3!} = 10$.

How do you calculate nPr?

Use $P(n, r) = \frac{n!}{(n - r)!}$. For P(5, 2): $\frac{5!}{3!} = 20$.

What is C(52, 5) used for?

It is the number of 5-card poker hands from a 52-card deck: 2,598,960.