Combination Calculator

When you’re counting how many ways to select a group of items without worrying about order, you’re dealing with combinations. When order matters, it’s permutations. These two concepts are the foundation of combinatorics, appearing in everything from probability problems and lottery calculations to password generation and team selection. But calculating large factorial values manually is tedious especially when dealing with numbers like 52 choose 5 or 100 choose 10.

This combination calculator handles both nCr (combinations) and nPr (permutations) with ease. Enter n and k, choose your calculation type and the tool delivers the exact result using big integer arithmetic. Whether you’re a statistics student learning about combinatorics, a data analyst calculating probabilities, or a gambler analyzing lottery odds, this calculator provides accurate results instantly. All calculations run locally in your browser, keeping your data private.

How to Use the Combination Calculator

1. Choose your calculation type: Combination (nCr) or Permutation (nPr) using the radio buttons.

2. Enter the total number of items using the first input field, this is the size of your full set.

3. Enter the number of items to choose using the second input field, this is the size of your selection.

4. Click Calculate or simply wait for automatic updates, the result appears instantly.

5. Review the result in the highlighted box, along with the formula displayed below.

6. Use the Copy button to save results or Share to send them to others.

How the Combination Calculator Formula Works

The calculator applies the standard formulas for combinations and permutations, using big integer arithmetic to handle large values without loss of precision.

Formula: nCr = n! / (r! × (n−r)!)

Combinations count the number of ways to choose r items from a set of n items where order does not matter. The formula uses factorials: n! is the product of all integers from 1 to n.

Formula: nPr = n! / (n−r)!

Permutations count the number of ways to arrange r items selected from a set of n items where order does matter. This is the number of ordered selections.

For example, to calculate 10C3: 10! / (3! × 7!) = 120. To calculate 10P3: 10! / 7! = 720. The calculator uses efficient algorithms that avoid computing full factorials for large n, reducing the risk of overflow while maintaining exact results.

Worked Example: Choosing a Committee

Consider a club with 10 members that needs to elect a president, vice-president, and secretary (3 positions). How many ways can the officers be chosen, and how many different committees of 3 members could be formed?

Step 1: Identify the known values

• n = 10 members

• k = 3 positions

Step 2: Permutations (order matters: president, VP, secretary are different roles)
nPk = 10P3 = 10! / (10−3)! = 10 × 9 × 8 = 720

Step 3: Combinations (order doesn’t matter: just 3 members)
nCk = 10C3 = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120

Interpretation: There are 720 different ways to fill the three officer positions (where order matters), and 120 different committees of 3 members (where order doesn’t matter). The difference between combinations and permutations is whether the arrangement order matters.