Combination Calculator

What is a Combination?

A combination is a way of selecting items from a collection where the order of selection does not matter. In combinations, unlike permutations, selecting A then B is considered the same as selecting B then A. The combination formula, also known as the binomial coefficient, calculates how many ways you can choose r items from n items.

The mathematical notation for combinations is C(n,r), nCr, or "n choose r". This appears frequently in probability theory, statistics, and combinatorics. For example, if you want to know how many different 5-card poker hands can be dealt from a 52-card deck, you would calculate C(52,5).

Combination Formula

The combination formula is: C(n,r) = n! / (r! × (n-r)!)

Where:

• n is the total number of items
• r is the number of items being chosen
• ! denotes factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1)

An important constraint is that r must be less than or equal to n (r ≤ n), since you cannot choose more items than are available.

How to Use This Calculator

1. Enter n: Input the total number of items in your set.
2. Enter r: Input the number of items you want to choose from the set.
3. Calculate: Click the "Calculate C(n,r)" button to get your result.
4. View formula: The calculator displays the complete formula with your values substituted in, showing exactly how the calculation works.

Common Examples

Lottery numbers: Calculate C(49,6) to find how many possible combinations exist when choosing 6 numbers from 49.

Team selection: If you have 15 players and need to choose 11 for a starting lineup where positions don't matter, calculate C(15,11).

Committee formation: To form a 3-person committee from 10 candidates, calculate C(10,3) = 120 possible committees.

Combinations vs Permutations

The key difference between combinations and permutations is that combinations don't care about order, while permutations do. When choosing 2 letters from {A,B,C}, there are 3 combinations: AB, AC, BC. But there are 6 permutations: AB, BA, AC, CA, BC, CB. Use combinations when the order doesn't matter (selecting team members) and permutations when order matters (arranging people in a line).

Properties of Combinations

Symmetry: C(n,r) = C(n,n-r). Choosing r items is the same as choosing which n-r items to leave out.

Edge cases: C(n,0) = 1 (one way to choose nothing), C(n,n) = 1 (one way to choose everything), C(n,1) = n.

Pascal's triangle: Each number in Pascal's triangle represents a combination: the nth row, rth position equals C(n,r).

When order matters, use the Permutation Calculator instead. The Factorial Calculator computes the n! values used in combination formulas. See the Probability Calculator for related probability calculations.