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Combinations (nCr)

Combinations Calculator

C(n,r) — order does not matter

What is Combinations (nCr)?

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Ever had to pick a small team from a big group of friends and wondered how many different ways you could do it? That is exactly what a 'combination' is. It is just a fancy way of counting groups where the order doesn't matter. If you are picking three friends to help you move, it doesn't matter if you call Sarah first or Mike first—they are both coming over to help. Because the result is the same group regardless of who you ask first, we use combinations to figure out the total number of possibilities. This calculator takes the headache out of the math for you. Instead of trying to scribble out long equations on a napkin, you just plug in the total number of items you have and how many you want to pick. It’s perfect for planning a fantasy football draft, deciding which three appetizers to order for the table, or even figuring out how many ways you can combine paint colors for a room makeover. It’s all about helping you understand your options without getting bogged down in the heavy lifting of factorials. Ultimately, this tool helps you make sense of large numbers by breaking down complex selection problems into a simple, reliable answer. Whether you are a student tackling a statistics homework assignment or just a curious person trying to figure out the odds of a game, this tool gives you the 'how many' without the guesswork. It turns a potentially confusing counting problem into a quick, practical insight so you can get back to your actual decision-making.

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Formula

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f(x)The math behind the magic is: C(n,r) = n! / (r! * (n-r)!). In plain English, you take the total number (n) and multiply it down to 1, then divide that by the number you chose (r) multiplied down to 1, times the remainder (n minus r) multiplied down to 1. The '!' symbol just means 'factorial', which is a fancy way of saying multiply all numbers from that value down to 1.

Variable Legend

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SymbolNameUnitDescription
nTotal items—The total number of things you have to choose from.
rItems chosen—The exact number of things you want to pick from the total pool.
C(n,r)Combination count—The final number of unique groups you can form.

How to Combinations (nCr)

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  1. 1Start by identifying your 'n', which is the total pool of items or people you have to choose from.
  2. 2Decide on your 'r', which is the specific number of items you actually want to pick for your group.
  3. 3Ask yourself: 'Does the order matter?' If it doesn't, you are in the right place to use this calculator.
  4. 4Input your numbers into the fields and hit calculate; the tool handles the heavy math, like factorials, instantly.
  5. 5Look at your result to see the total number of unique combinations possible.

Worked Examples

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Example 1Choosing dinner side dishes
Given:Pick 2 sides from a menu of 6
Result:C(6,2) = 15

A great way to see how many meal combos you have.

If you have 6 sides and want to pick 2, there are 15 different pairs you can make. It doesn't matter if you get the fries then the salad, or the salad then the fries—the final plate is the same.

Example 2Starting lineup for a pickup game
Given:Pick 5 players from a group of 10
Result:C(10,5) = 252

Perfect for organizing sports teams.

With 10 friends showing up to the park, you can create 252 different 5-person teams. Since you are just picking a group to play, the order of picking doesn't create a different team.

Example 3Vacation souvenir shopping
Given:Pick 3 postcards from a rack of 15
Result:C(15,3) = 455

See how many sets of gifts you can grab.

You have 15 beautiful designs to choose from and you want to buy 3. The calculator shows there are 455 unique sets of three you could walk away with.

Example 4Office book club
Given:Pick 4 books from a list of 8 suggestions
Result:C(8,4) = 70

Planning your reading list for the year.

If your team suggests 8 books but you only have time to read 4, there are 70 different ways to curate your final reading list for the year.

Real-World Applications

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Helping event planners figure out how many different seating arrangements or menu combinations they can offer guests.

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Assisting sports coaches in calculating how many unique starting lineups they can build from their roster.

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Supporting students in learning the basics of probability for math class or test prep.

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Helping hobbyists calculate the variety of color palettes or material combinations for their DIY home projects.

Special Cases

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Choosing none or everything

If you choose 0 items or all of the items, the result is always 1. It makes sense—there is only one way to pick nothing, and only one way to pick the entire group!

When 'r' is larger than 'n'

If you try to pick more items than you actually have in your pile, the result will be 0. You simply cannot select 5 apples if you only have 3 in your basket.

Quick Look at Possible Groups

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Total Pool (n)Selection Size (r)Possible GroupsReal-world context
5210Pairs from 5 items
103120Trios from 10 items
124495Quartets from 12 items
5252598960Poker hands

Frequently Asked Questions

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Q

Why do I keep getting different answers when I swap my numbers?

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It is because n and r play different roles! The 'n' is your total pile of stuff, and 'r' is just the small handful you are grabbing. If you flip them, you are asking a completely different question, so the math will naturally give you a different result.

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Does it matter if I pick the same item twice?

A

This specific calculator assumes you are picking distinct items without putting them back. If you need to pick the same item multiple times, you would need a different type of calculation called 'combinations with replacement'.

Q

How do I know if I should use this or a permutation calculator?

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Just ask yourself if the order of your items changes the outcome. If you are making a fruit salad, order doesn't matter (that's a combination!). If you are awarding 1st, 2nd, and 3rd place trophies, order definitely matters (that's a permutation).

Q

What if my result is a massive number?

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That is totally normal! Combinations grow very fast as you add more items to your pool. Don't worry, the calculator is designed to handle those huge numbers so you don't have to.

Q

Can I use this for my school project?

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Absolutely! This is a great way to double-check your work after you have tried solving a probability or statistics problem by hand. It helps you see if your logic is on the right track.

Q

Why is the result 1 when I choose '0' or 'all'?

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Think about it like this: there is only one way to pick nothing at all (you just don't pick anything), and there is only one way to pick every single item in the pile (you just take the whole thing). The math reflects that reality!

Q

Is there a trick to doing this quickly in my head?

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Yes! Remember that C(n,r) is the same as C(n, n-r). If you are picking 8 items out of 10, it is the same math as picking 2 items out of 10 to leave behind. Choosing the smaller number to calculate usually makes the mental math much easier.

Common Mistakes to Avoid

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  • !Mixing up n (the total) and r (the selection).
  • !Trying to use this when the order of items actually matters (use a Permutation calculator instead!).
  • !Forgetting that you can't pick more items than you have available.
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Pro Tip

If the numbers look too big to manage, remember the symmetry rule! Picking 98 items out of 100 is the exact same as picking the 2 items you're going to leave behind. It makes the math so much faster.

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Did you know?

Did you know that the 'lottery' isn't just luck? Every time you buy a ticket, you are participating in a massive combinations problem. For a standard 6-number lottery out of 49, there are over 13 million possible combinations, which is why your odds of hitting the jackpot are so slim!

📖Difficulty:Beginner

Variable Legend

n= total itemsr= items chosenC(n,r)= number of combinations

Combinations formula

Order does not matter.

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Pascal's triangle identity

Each value is the sum of the two above it.

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Symmetry

Choosing r is the same as leaving out n−r.

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Combinations with repetition

Allow repeating items.

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Related Articles
→

How to Calculate Probability: Formulas, Examples, and Real Applications

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→

How to Calculate Permutations and Combinations

7 min read

Compare With
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Combinations vs Permutations

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Factorial vs Combinations

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Reviewed October 2026
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