math calculator

Combination Calculator — Calculate nCr & n Choose k Instantly

Calculate combinations (nCr) instantly with step-by-step work, with or without repetition. Perfect for lottery odds, poker hands, team selection, and sampling.

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Select Mode

Inputs

Total Items (n)

Items to Choose (r)

Allow repetition of items

Results

Combinations Result

792

Related Permutations (nPr)95,040

Combination Step-by-Step

12C5 = 12! / (5!(12 - 5)!)

= (12 × 11 × 10 × 9 × 8) / 5!

= 95,040 / 120

= 792

Factorials Reference

n! (12!) = 479,001,600

r! (5!) = 120

(n - r)! (7!) = 5,040

Bookmark this combination calculator

Save this page for quick access when working on combinatorics homework, probability problems, exam prep, or real-world planning like lottery odds and team selection.

Formula

Combination Formula

Without repetition:
  nCr = n! / (r!(n - r)!)

With repetition:
  nCr (rep) = C(n + r - 1, r)

Related:
  nCr = nPr / r!

Combinations count unordered selections — only which items are chosen matters, not the order they were picked in. Every combination corresponds to r! different permutations, which is why nCr is always smaller than nPr.

Step by step

How the calculation works

1Enter the total number of items (n) and how many you're choosing (r).
2Toggle "Allow repetition" on if items can be chosen more than once (like ice cream scoops); leave it off for problems like lottery numbers or committee members.
3Review the highlighted combinations result, plus the related permutations (nPr) shown for reference.
4Check the step-by-step factorial breakdown, the factorials reference card, and (for small sets) the full list of generated combinations.
5Switch to the Real-World Scenarios tab to explore the same math applied to lotteries, poker hands, team selection, sampling, food toppings, and repetition.

Example

Combinatorics Example — Choosing 3 Committee Members from 9

Choosing 3 members from 9 candidates for a committee: Combinations = 9! / (3! × 6!) = 84 ways — the order they're picked in doesn't matter. If instead you were assigning those 3 picks to President, Vice President, and Secretary, order would matter and you'd need Permutations = 9! / 6! = 504 ways instead.

Related guides

Learn more about this calculation

Guide

Standard Deviation Guide: Measure Data Spread Clearly

Understand variance, population vs sample formulas, and data spread with worked examples tied to the live DTC standard deviation calculator.

Guide

Random Number Generation Guide: Seeded PRNGs, True Randomness, and Safe Use Cases

Learn the difference between deterministic seeded generators, browser pseudo-random functions, and cryptographically strong randomness so you can use the right tool for the job.

Permutation

Permutation Calculator

Calculate permutations (nPr) for problems where order matters — passwords, seating, and rankings.

Factorial

Factorial & Gamma Calculator

Calculate factorials, double factorials, and the gamma function for any input.

Number Generator

Random Number Calculator

Generate random numbers within a custom range, with or without repeats.

Statistics

Descriptive Statistics Calculator

Compute mean, median, mode, variance, and standard deviation for a dataset.

math calculator

Combination Calculator — Calculate nCr & n Choose k Instantly

Calculate combinations (nCr) instantly with step-by-step work, with or without repetition. Perfect for lottery odds, poker hands, team selection, and sampling.

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Select Mode

Inputs

Total Items (n)

Items to Choose (r)

Allow repetition of items

Results

Combinations Result

792

Related Permutations (nPr)95,040

Combination Step-by-Step

12C5 = 12! / (5!(12 - 5)!)

= (12 × 11 × 10 × 9 × 8) / 5!

= 95,040 / 120

= 792

Factorials Reference

n! (12!) = 479,001,600

r! (5!) = 120

(n - r)! (7!) = 5,040

Bookmark this combination calculator

Save this page for quick access when working on combinatorics homework, probability problems, exam prep, or real-world planning like lottery odds and team selection.

Summary

Key output metrics

Combinations (nCr)

792

Combinations without repetition.

Total Items (n)

Chosen (r)

Related Permutations (nPr)

95,040

D3 visualization

Combinations vs Permutations Growth

Combinations (unordered choices) grow much more slowly than permutations (ordered choices) as the selection size increases.

Guide

How to understand and use this calculator

Quick Answer

What is a combination? A combination is the number of ways to choose r items from a set of n distinct items where order does not matter.

Formula:

nCr = n! / (r!(n - r)!)

Simple example: How many ways to choose 3 cards from a deck of 9?

9C3 = 9! / (3! × 6!) = 9×8×7 / (3×2×1) = 84

Key difference:

Combinations: Order doesn't matter (selections)
Permutations: Order matters (arrangements)

Quick reference:

5C2 = 10
10C3 = 120
52C5 = 2,598,960 (poker hands)

Use the Combination Calculator

To use the combinations calculator:

1. Enter the total number of items (n).

2. Enter the number of items to choose (r).

3. Toggle repetition on if items can be chosen more than once.

4. Review the number of combinations (nCr) and the step-by-step work shown below.

Example input: Total Items (n) = 12, Items to Choose (r) = 5.

Output:

Combinations (12C5) = 792
Step-by-step work: 12C5 = 12!/(5!×7!) = 792

Another example: Total Items (n) = 5, Items to Choose (r) = 2.

Combinations (5C2) = 10
Step-by-step work: 5C2 = 5!/(2!×3!) = (5×4)/(2×1) = 10

What is a Combination?

A combination is the number of ways to select a group of items from a larger set where the order of selection does not matter.

Think of it like this: if you're choosing a team of 3 people from 10 candidates, the order in which you pick them doesn't matter. Alice, Bob, and Charlie is the same team as Charlie, Bob, and Alice.

Key characteristics:

Items are distinct (different from each other)
Order does not matter (selections, not arrangements)
No repetition allowed (usually — toggle repetition on for cases where it is)

Simple example: How many ways to choose 2 items from {A, B, C}?

Combinations: AB, AC, BC = 3 ways
Notice: AB is the same as BA!

The combination formula:

nCr = n! / (r!(n - r)!)
Where n = total items, r = items to choose

Combination vs Permutation — What's the Difference?

Feature comparison:

Order: Combinations — order doesn't matter. Permutations — order matters.
Also called: Combinations — selections. Permutations — arrangements.
Example: Combinations — AB = BA. Permutations — AB ≠ BA.
Formula: Combinations — n! / (r!(n-r)!). Permutations — n! / (n-r)!.
Value: Combinations are always smaller. Permutations are always larger.
Used for: Combinations — lottery, teams, sampling. Permutations — passwords, seating, rankings.

Example: Set {A, B, C}, choose 2.

Combinations (order doesn't matter): AB, AC, BC = 3
Permutations (order matters): AB, AC, BA, BC, CA, CB = 6

When to use which:

Lottery → Combinations. Order of numbers doesn't matter.
Choosing a team → Combinations. Who's on the team, not their order.
Picking flavors → Combinations. Order of flavors doesn't matter.
Passwords → Permutations. Order of characters matters.
Seating arrangements → Permutations. Where people sit matters.
Race rankings → Permutations. First, second, third matters.

Need the arrangement version instead? Try the Permutation Calculator.

The Combination Formula (nCr)

The nCr combination formula calculates the number of ways to choose r items from n distinct items.

Formula:

nCr = n! / (r!(n - r)!)

Where:

n = total number of items
r = number of items to choose
! = factorial (multiply all numbers from 1 to that number)

Expanded form:

nCr = [n × (n-1) × (n-2) × ... × (n - r + 1)] / [r × (r-1) × ... × 1]

Example: 9C3 = (9 × 8 × 7) / (3 × 2 × 1) = 504 / 6 = 84

Special cases:

nC0 = 1 (choose nothing)
nC1 = n (choose one item)
nCn = 1 (choose all items)
0! = 1 (mathematical convention)

How to Calculate Combinations — Step by Step

Calculating combinations is simple when you follow these steps.

Step 1: Identify n (total items) and r (items to choose). Example: 12 items, choose 5 → n = 12, r = 5.

Step 2: Write the formula: nCr = n! / (r!(n - r)!)

Step 3: Plug in the numbers: 12C5 = 12! / (5! × 7!)

Step 4: Expand and simplify: 12C5 = (12×11×10×9×8) / (5×4×3×2×1) = 95,040 / 120 = 792.

Answer: 12C5 = 792

More examples:

5C2 = 5!/(2!×3!) = (5×4)/(2×1) = 10
6C3 = 6!/(3!×3!) = (6×5×4)/(3×2×1) = 20
10C4 = 10!/(4!×6!) = (10×9×8×7)/(4×3×2×1) = 210
7C2 = 7!/(2!×5!) = (7×6)/(2×1) = 21
8C3 = 8!/(3!×5!) = (8×7×6)/(3×2×1) = 56

How to Use This as a Combinations Calculator

Our combinations calculator can be used for many practical purposes — try each one live in the Real-World Scenarios tab above.

1. Lottery and Gambling:

Pick 6 from 49 → 49C6 = 13,983,816
Pick 5 from 69 + Powerball (1 from 26) → 69C5 × 26 = 292,201,338
Pick 5 from 70 + Mega Ball (1 from 25) → 70C5 × 25 = 302,575,350

2. Poker Hands:

5-card poker → 52C5 = 2,598,960
Texas Hold'em starting hands → 52C2 = 1,326

3. Choosing Teams:

Choose 3 from 10 for a committee → 10C3 = 120
Choose 5 from 20 for a basketball team → 20C5 = 15,504

4. Sampling and Statistics:

Choose 10 from 100 for a survey → 100C10 = 17,310,309,456,440

5. Food and Flavors:

Choose 3 toppings from 15 → 15C3 = 455

Combinations with Repetition

Sometimes you can choose the same item more than once. This is called combinations with repetition.

Formula:

C(n+r-1, r) = (n+r-1)! / (r!(n-1)!)

Example: You have 3 flavors of ice cream (chocolate, vanilla, strawberry) and you want to choose 2 scoops. You can choose the same flavor twice.

n = 3, r = 2
C(3+2-1, 2) = C(4, 2) = 4! / (2! × 2!) = 6

Possible combinations: CC, CV, CS, VV, VS, SS = 6

Toggle "Allow repetition" in the calculator above, or try the "Combinations with Repetition" scenario, to explore this interactively.

When to use repetition:

Lottery numbers → Without. The same number can't appear twice.
Ice cream scoops → With. You can have two scoops of chocolate.
Choosing team members → Without. The same person can't be chosen twice.
Selecting toppings → Without (usually).
Password characters → With. You can use the same character twice (though passwords are really a permutation problem — see the Permutation Calculator).

Real-World Applications of Combinations

Lottery. 6 numbers from 49 → 13,983,816 combinations. Why it matters: understanding the odds of winning.
Poker. 5 cards from 52 → 2,598,960 combinations. Why it matters: probability of specific hands.
Team Selection. 3 from 10 → 120 combinations. Why it matters: choosing committees or teams.
Sampling. 10 from 100 → over 17 trillion combinations. Why it matters: statistical analysis.
Food. 3 toppings from 15 → 455 combinations. Why it matters: menu planning.
Sports. 5 starters from 12 → 792 combinations. Why it matters: team selection.
Cryptography. Password character combinations. Why it matters: security analysis.
Genetics. Gene combinations. Why it matters: inheritance studies.

Practice Problems

Problem 1: How many ways to choose 2 flavors from 7 ice cream flavors? Solution: n = 7, r = 2 → 7C2 = (7 × 6) / (2 × 1) = 21.

Problem 2: How many 3-person committees can be formed from 12 people? Solution: n = 12, r = 3 → 12C3 = (12 × 11 × 10) / (3 × 2 × 1) = 220.

Problem 3: How many different 5-card poker hands are possible from a standard 52-card deck? Solution: n = 52, r = 5 → 52C5 = (52×51×50×49×48) / (5×4×3×2×1) = 2,598,960.

Problem 4: A pizza shop offers 8 toppings. How many ways to choose 4 toppings? Solution: n = 8, r = 4 → 8C4 = (8×7×6×5) / (4×3×2×1) = 70.

Problem 5: How many ways to choose a committee of 2 people from 10? Solution: n = 10, r = 2 → 10C2 = (10 × 9) / (2 × 1) = 45.

Answers: 1) 21 — 2) 220 — 3) 2,598,960 — 4) 70 — 5) 45

Common Combination Mistakes

Using combinations when order matters. Mistake: using nCr for passwords or rankings. Fix: ask "Does order matter?" If yes → use permutations (nPr).
Using permutations when order doesn't matter. Mistake: using nPr for lottery or team selection. Fix: if order doesn't matter → use combinations (nCr).
Forgetting to divide by r! in the denominator. Mistake: nCr = n! / (n-r)! (wrong — that's permutations!). Fix: nCr = n! / (r!(n-r)!)
Confusing n and r. Mistake: 9C3 = 3C9 (nonsense). Fix: n is the total, r is the number chosen (r ≤ n).
Not simplifying the fraction. Mistake: leaving 9C3 as 504/6. Fix: simplify to 84.
Forgetting that r can be 0. Mistake: nC0 = 0 (wrong). Fix: nC0 = 1 (choose nothing, 1 way).
Misreading the problem. Mistake: using the wrong numbers for n or r. Fix: carefully read the problem to identify n and r.
Not checking answer reasonableness. Mistake: getting an answer that seems too large or too small. Fix: does the answer make sense? If not, double-check.

What This Calculator Can and Cannot Do

This calculator CAN help with:

Calculating combinations (nCr) and understanding n choose k problems
Combinations with and without repetition
Step-by-step work shown for every calculation, plus the related permutation (nPr) for reference
Real-world applications — lottery, poker, teams, sampling, and food — in the Scenarios tab
Generating the full list of combinations for small sets (n and r of 5 or less)
Both small and very large numbers, with results that stay accurate even past standard number limits

This calculator CANNOT do:

Calculate permutations as the primary output — use the Permutation Calculator when order matters
Calculate probabilities directly
Handle combinations of sets with repeated, indistinguishable items using the standard nCr formula (that requires a multinomial approach)
Replace understanding of the underlying concepts

Final Summary

A combination calculator is essential for anyone working with selections and probabilities — whether you're a student learning combinatorics, a teacher preparing lessons, or someone trying to understand lottery odds.

Remember the key formulas:

Combinations (nCr): n! / (r!(n-r)!) — order doesn't matter
Permutations (nPr): n! / (n-r)! — order matters
Relationship: nCr = nPr / r!

Quick reference:

5C2 = 10 (selections)
5P2 = 20 (arrangements)
10C3 = 120
52C5 = 2,598,960

Start calculating now! Just enter your values above and get instant results — completely free.

Related concepts

Binomial Coefficient

Another name for nCr — the number of ways to choose r items from n, and the coefficients in a binomial expansion.

Stars and Bars

The counting technique behind combinations with repetition, used to derive C(n+r-1, r).

Combinatorics

The branch of mathematics concerned with counting, arranging, and combining objects, underlying probability and statistics.

Frequently Asked Questions

What is a combination calculator?+

A combination calculator finds the number of ways to choose r items from n distinct items where order doesn't matter.

What is the combination formula?+

The combination formula is nCr = n! / (r!(n-r)!). This is also known as the nCr combination formula.

What is the difference between combination and permutation?+

In combinations, order doesn't matter (selections). In permutations, order matters (arrangements).

When do I use combinations?+

Use combinations when the order of items doesn't matter, such as lottery numbers, choosing a team, or picking toppings.

What is n choose k?+

"n choose k" is another name for combinations. It means choosing k items from n items.

What is the relationship between combinations and permutations?+

nCr = nPr / r!. Combinations are smaller because order doesn't matter.

How do I calculate combinations?+

To calculate combinations: identify n and r, use the formula nCr = n!/(r!(n-r)!), and calculate.

What is 0!?+

0! = 1 (mathematical convention).

Can I use combinations with repetition?+

Yes — toggle "Allow repetition" in the calculator. The formula becomes C(n+r-1, r) = (n+r-1)! / (r!(n-1)!).

What is the difference between nCr and nPr?+

nCr (combinations) is for selections where order doesn't matter. nPr (permutations) is for arrangements where order matters.

How many ways to choose 3 from 10?+

10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120 ways.

How many 5-card poker hands are possible?+

52C5 = 2,598,960 possible poker hands.

How many ways to choose 2 toppings from 8?+

8C2 = (8 × 7) / (2 × 1) = 28 ways.

What is a binomial coefficient?+

The binomial coefficient is another name for combinations. It's the number of ways to choose r items from n items.

Is this combination calculator free?+

Yes — completely free with no registration required.

Formula

Combination Formula

Without repetition:
  nCr = n! / (r!(n - r)!)

With repetition:
  nCr (rep) = C(n + r - 1, r)

Related:
  nCr = nPr / r!

Step by step

How the calculation works

1Enter the total number of items (n) and how many you're choosing (r).
2Toggle "Allow repetition" on if items can be chosen more than once (like ice cream scoops); leave it off for problems like lottery numbers or committee members.
3Review the highlighted combinations result, plus the related permutations (nPr) shown for reference.
4Check the step-by-step factorial breakdown, the factorials reference card, and (for small sets) the full list of generated combinations.
5Switch to the Real-World Scenarios tab to explore the same math applied to lotteries, poker hands, team selection, sampling, food toppings, and repetition.

Example

Combinatorics Example — Choosing 3 Committee Members from 9

Related guides

Learn more about this calculation

Guide