Combinations Calculator

How the Combinations Calculator works

The Combinations Calculator computes both nCr (combinations, where order does not matter) and nPr (permutations, where order matters) for any non-negative integers n and r. Enter the total number of items (n) and how many you want to choose (r), and the calculator returns both values along with the factorial computations used in the calculation.

The formulas

Combinations (nCr) — order does not matter

nCr = n! ÷ ( r! × (n − r)! )

"How many ways can I choose r items from n items, if order does not matter?"

Permutations (nPr) — order matters

nPr = n! ÷ (n − r)!

"How many ways can I arrange r items from n items, if order matters?"

Factorial

n! = n × (n−1) × (n−2) × ... × 2 × 1, with 0! = 1

Worked examples

Combinations: How many ways to choose 3 students from a class of 10? C(10, 3) = 10! ÷ (3! × 7!) = 3628800 ÷ (6 × 5040) = 120.

Permutations: How many ways to arrange 3 students in a line from a class of 10? P(10, 3) = 10! ÷ 7! = 3628800 ÷ 5040 = 720.

Notice that P(10, 3) = C(10, 3) × 3! = 120 × 6 = 720. This makes sense: for each combination of 3 students, there are 3! = 6 ways to arrange them in a line.

When to use which

The choice between combinations and permutations hinges on one question: does order matter? If yes, use permutations. If no, use combinations. Some examples:

• Combinations: Selecting a committee (a team of {A, B, C} is the same as {B, A, C}). Picking lottery numbers (1, 7, 13 wins the same prize regardless of order). Choosing pizza toppings.
• Permutations: Race results (1st, 2nd, 3rd are different). Arranging books on a shelf. Creating passwords (ABCD is different from DCBA). Listing phone numbers.

A useful test: ask yourself "if I swap two of the chosen items, is it a different result?" If yes (race positions, passwords, shelf arrangements), use permutations. If no (committees, toppings, lottery picks), use combinations. Getting this distinction right is essential — using the wrong one gives an answer that is off by a factor of r!, which can be enormous.

Factorial growth

Factorials grow astonishingly fast. 5! = 120, 10! = 3,628,800, 20! = 2.4 × 10¹⁸, and 70! exceeds the number of atoms in the observable universe. This is why brute-force algorithms that try every permutation of even 20 items are computationally infeasible — and why clever algorithms (sorting, searching, optimization) are so valuable. The factorial\'s explosive growth also explains why large random samples behave so predictably: with 100 items, the number of possible orderings is so vast that almost any specific arrangement is vanishingly unlikely, which is the foundation of statistical reasoning.

Frequently asked questions