Combination Calculator

In just a few minutes, you’ll master combination fundamentals, understand the difference between combinations and permutations, and learn practical applications that you can apply immediately to your probability and statistical analysis work.

How to Use the Combination Calculator

Calculating combinations requires different approaches for different mathematical scenarios. Carefully read the instructions below and decide which method is best for your situation:

Take a look at your data:

Are you selecting items where order doesn’t matter?

• If you’re choosing a team, committee, or group, use the combination (nCr) method.
• If you need to find how many ways to select r items from n total, use binomial coefficient calculation.
• If order is irrelevant in your selection process, combinations are the correct choice.

Are you arranging items where order matters?

• Use the permutation (nPr) method for arrangements, rankings, or sequences.
• If you’re assigning positions or creating ordered lists, use permutation calculations.
• When the arrangement sequence is important, permutations provide the answer.

Are you analyzing probability or odds?

• Use selection probability mode to calculate likelihood of specific outcomes.
• Apply combinations to lottery odds and gaming probability calculations.
• Combine with probability theory for statistical inference.

Combination and Permutation Formulas

Combination Formula — selecting without order

Use this when calculating the number of ways to choose r items from n total items where order doesn’t matter.

Where:

• C(n,r) — Combination result (also written as nCr or ⁿCᵣ)
• n — Total number of items available
• r — Number of items to select
• ! — Factorial notation (n! = n × (n-1) × … × 1)

Permutation Formula — arranging with order

Use this when calculating the number of ways to arrange r items from n total items where order matters.

Where:

• P(n,r) — Permutation result (also written as nPr or ⁿPᵣ)
• n — Total number of items available
• r — Number of items to arrange

Binomial Coefficient — alternative notation

Where:

• (n choose r) — Binomial coefficient notation
• Same calculation as C(n,r) — Different notation for combinations

Relationship Formula — connecting combinations and permutations

This shows that permutations equal combinations multiplied by the arrangements of selected items.

Combination Calculation Example

Let’s walk through a real example. Imagine you’re forming a committee of 3 people from a group of 10 candidates:

Step 1: Identify the values

• Total candidates (n): 10 people
• Committee size (r): 3 people
• Order matters: No (it’s a committee, not ranked positions)
• Formula to use: C(10,3) = 10! / (3! × 7!)

Step 2: Calculate the factorials

• 10! = 10 × 9 × 8 × 7! (we can cancel the 7!)
• 3! = 3 × 2 × 1 = 6
• Simplified: (10 × 9 × 8) / (3 × 2 × 1)

Step 3: Complete the calculation

• Numerator: 10 × 9 × 8 = 720
• Denominator: 3! = 6
• Result: 720 / 6 = 120

Final result: There are 120 different ways to choose 3 people from a group of 10 for a committee.

Combinations vs. Permutations vs. Arrangements

People often wonder about the difference between these combinatorial concepts. Here’s a simple way to think about it:

• Combinations: Selection where order doesn’t matter (choosing team members)
• Permutations: Arrangement where order matters (assigning positions)
• Arrangements: General term for organizing items in sequence
• Variations: Alternative term for permutations in some contexts

When to use each:

• Use combinations for committees, teams, groups, or any unordered selection
• Use permutations for rankings, sequences, passwords, or ordered arrangements
• Use probability calculations when determining likelihood of specific outcomes
• Use both together for complex scenarios involving selection and arrangement

Mathematical Context and Applications

Here’s some background that might interest you: combinations are fundamental in discrete mathematics, probability theory, statistics, genetics, computer science, and optimization across various scientific and practical domains.

Combinatorial analysis provides the mathematical foundation for calculating probabilities in games of chance, analyzing genetic inheritance patterns, determining sample sizes in statistics, and solving optimization problems in computer algorithms.

Practical applications include lottery odds calculation, quality control sampling, tournament bracket design, password security analysis, and machine learning feature selection. The principles also appear in chemistry (molecular combinations), biology (genetic variations), and economics (portfolio selection).

This is why combination analysis matters in your situation: proper combinatorial understanding ensures accurate probability calculations, efficient algorithm design, and informed decision-making in scenarios involving selection and arrangement.

Frequently Asked Questions

What would I get if I choose 3 items from 8 total items?

Using the combination formula C(8,3) = 8!/(3! × 5!) = (8 × 7 × 6)/(3 × 2 × 1) = 336/6 = 56. There are 56 different ways to choose 3 items from 8 when order doesn’t matter.

How do I know whether to use combinations or permutations?

Ask yourself: “Does order matter?” If you’re forming a team or selecting a group where the order of selection is irrelevant, use combinations. If you’re creating a ranking, assigning positions, or the sequence matters, use permutations. Combinations are always smaller than or equal to permutations for the same n and r.

What if r is larger than n in my calculation?

Mathematically, C(n,r) = 0 when r > n because you cannot select more items than are available. Similarly, P(n,r) = 0 when r > n. Most calculators will show an error or zero result. Check your input values to ensure r ≤ n.

Can I calculate combinations for very large numbers?

Yes, but results may be extremely large. Our calculator handles values up to reasonable computational limits and displays results in scientific notation when needed. For very large calculations, consider using logarithmic approaches or specialized software for precise results.

How do I interpret C(n,0) and C(n,n)?

C(n,0) = 1 (there’s exactly one way to choose nothing), and C(n,n) = 1 (there’s exactly one way to choose everything). These boundary cases are important in mathematical proofs and probability calculations, particularly in binomial theorem applications.

What’s the relationship between combinations and the binomial theorem?

Binomial coefficients C(n,r) appear in the binomial theorem: (x+y)ⁿ = Σ C(n,r) × xʳ × y^(n-r). Each coefficient represents the number of ways to choose r factors of x from n total factors in the expansion.

How do I calculate the probability of specific combinations?

Probability = (Favorable combinations) / (Total possible combinations). For example, if you want exactly 2 red balls when drawing 5 balls from a mix of 10 red and 15 blue balls: P = [C(10,2) × C(15,3)] / C(25,5).

Can combinations be used with replacement?

The standard combination formula assumes selection without replacement. For selection with replacement (where the same item can be chosen multiple times), use the formula: C(n+r-1,r) = (n+r-1)! / (r! × (n-1)!). This is called “combinations with repetition.”

What are some common mistakes in combination calculations?

Common errors include: confusing combinations with permutations, forgetting that order doesn’t matter in combinations, using the wrong formula for large numbers, and misunderstanding boundary cases like C(n,0). Always verify whether your problem involves ordered or unordered selection.

How do combinations relate to Pascal’s triangle?

Pascal’s triangle displays binomial coefficients where each entry equals C(n,r). Row n contains the coefficients C(n,0), C(n,1), …, C(n,n). Each number is the sum of the two numbers above it, demonstrating the identity C(n,r) = C(n-1,r-1) + C(n-1,r).