Statistics
Combination Calculator
Find how many ways you can choose r items from a set of n. Switch between combinations, permutations, and choices with repetition, see the working term by term, list the actual selections, and look up any value on Pascal's triangle.
Set n and r
Choose r from n — order ignored, no repeats.
The whole set
The size of each selection
nCr
5C2 =10
10 ways to choose 2 of 5 items when order doesn't matter.
Order ignored · no repeats
nCr
10
nPr
20
nPr ÷ nCr
2
Try an example
No sign-up · Free
The actual selections
10 totalItems labelled A–E. Each box is one combination — reorderings count as the same one.
ABACADAEBCBDBECDCEDE
The basics
What is a combination?
A combination is a selection where order is irrelevant. Picking a fruit salad of apple, banana, and cherry is the same choice however you say it — so the three items are one combination, not six. That single idea, “which items, not their order,” is what separates a combination from a permutation and gives it its own formula.
nCr = n! / (r! · (n − r)!)
Why we divide
Start from every ordering of the whole set, then divide out what doesn't count: the r! ways to arrange the items you picked and the (n − r)! ways to arrange the ones you left behind.
5C2 = 5! / (2!·3!) = 10
nPr = nCr × r!
Combinations vs permutations
A permutation cares about order, a combination doesn't. Every combination of r items can be shuffled r! ways, so there are always more permutations than combinations.
8P3 = 56 × 6 = 336
Method
How to calculate nCr by hand
You never need the full factorials. The short multiplicative form keeps the numbers small and exact.
- 1
Write the formula
Begin with the definition. The factorials in the denominator are the orderings you're throwing away.
nCr = n! ÷ (r! · (n − r)!) - 2
Cancel (n − r)!
The large (n − r)! divides straight into n!, leaving just the top r factors of n!.
5C2 = (5 × 4) ÷ (2 × 1) - 3
Multiply r terms over r!
Multiply the r descending numbers on top, then divide by r! on the bottom.
= 20 ÷ 2 - 4
Simplify
The quotient is always a whole number — that is your count of combinations.
= 10
Worked examples
See it on real problems
The same recipe handles a pizza order, a committee, and a lottery — only the numbers change.
Problem Choose 3 toppings from 8 available.
Set up: 8C3 = 8! ÷ (3! · 5!)
Cancel and multiply 3 terms: (8 × 7 × 6) ÷ (3 × 2 × 1)
8C3 = 56
The four cases
Order and repetition decide the formula
Counting problems split four ways depending on two yes/no questions: does order matter, and can items repeat? Pick the right cell and the formula follows.
Order matters · no repeats
Permutation — nPr
n! / (n − r)!
Race podiums: 8P3 = 336
Order matters · repeats allowed
Tuples
nʳ
PIN codes: 10³ = 1,000
Order ignored · no repeats
Combination — nCr
n! / (r!·(n − r)!)
Poker hands: 52C5 = 2,598,960
Order ignored · repeats allowed
Multiset — C(n+r−1, r)
(n + r − 1)! / (r!·(n − 1)!)
Ice-cream scoops: C(7,3) = 35
Where it shows up
Why combinations matter
Any time you count unordered selections — odds, samples, subsets — nCr is doing the work.
49C6
Lottery & gambling
Jackpot odds are one over the number of combinations of numbers. Poker hand rankings come straight from counting the ways to draw cards.
nCr pⁿ
Probability
The binomial distribution multiplies nCr by the chance of each outcome to model coin flips, defect rates, and clinical trials.
(x + y)ⁿ
Algebra & beyond
As binomial coefficients, nCr expands powers of a sum and seeds the rows of Pascal's triangle used across combinatorics.
Worth knowing
Identities worth remembering
A handful of clean rules make combinations faster to compute and easier to reason about.
nCr = nC(n − r)
Symmetry
Choosing what to take is the same as choosing what to leave, so the count is a mirror image across the middle of each row.
nC0 = nCn = 1
The edges are 1
There is exactly one way to choose nothing and one way to choose everything — the outer numbers of every Pascal row.
nCr = (n−1)C(r−1) + (n−1)Cr
Pascal's rule
Each value is the sum of the two above it, which is precisely how Pascal's triangle is built row by row.
Σ nCr = 2ⁿ
Row sum
Add up a whole row and you get a power of two — the total number of subsets of an n-element set.
nC1 = n
Choosing one
Picking a single item from n can be done in exactly n ways, no formula needed.
nPr = nCr × r!
Back to permutations
Multiply a combination count by r! and you recover the number of ordered arrangements.
Quick chart
Look it up on Pascal's triangle
Every entry in Pascal's triangle is a combination: row n, position r holds nCr. Tap any number to read its value and the (n, r) it answers.
5C2 =10row 5, position 2
n=0
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
n=11
n=12
n=13
n=14
Each number is the sum of the two above it — and equals nCr for that row and position. The edges are all 1 because nC0 and nCn are always 1.
Reference
A small nCr table
The most-looked-up values at a glance. Each cell is nCr for that row and column.
| n \ r | r=0 | r=1 | r=2 | r=3 | r=4 | r=5 |
|---|---|---|---|---|---|---|
| n=5 | 1 | 5 | 10 | 10 | 5 | 1 |
| n=6 | 1 | 6 | 15 | 20 | 15 | 6 |
| n=7 | 1 | 7 | 21 | 35 | 35 | 21 |
| n=8 | 1 | 8 | 28 | 56 | 70 | 56 |
| n=10 | 1 | 10 | 45 | 120 | 210 | 252 |
| n=12 | 1 | 12 | 66 | 220 | 495 | 792 |
Pitfalls
Where people slip up
Most wrong answers come from one of these three confusions — easy to avoid once you spot them.
order ≠ ignored
Using nCr when order matters
If swapping two picks gives a genuinely different outcome — first vs second place — you need a permutation, not a combination.
podium → use 8P3 = 336, not 8C3 = 56
r ≤ n only
Letting r exceed n
You can't choose more distinct items than exist. If r is bigger than n, either the count is zero or the problem really allows repetition.
without repeats, 5C8 has no meaning
repeats change the rule
Forgetting repetition
When the same item can be picked again, the plain nCr undercounts — switch to the multiset formula C(n + r − 1, r).
3 scoops of 5 flavours → C(7,3) = 35
FAQ
Common questions
A combination is a selection of items from a larger group where the order does not matter. Choosing apples, bananas, and cherries is the same combination as cherries, apples, and bananas — only which items are picked counts, not their arrangement.
A permutation counts arrangements where order matters; a combination counts selections where it does not. There are always at least as many permutations as combinations, because each combination of r items can be arranged in r! different orders. That is exactly the relationship nPr = nCr × r!.
The number of combinations of n items taken r at a time is nCr = n! / (r! · (n − r)!), where ! means factorial. The denominator divides out the r! orderings of the chosen items and the (n − r)! orderings of the items left behind.
Use the short multiplicative form: multiply the r largest numbers from n downward on top, divide by r! on the bottom. For 5 choose 2 that is (5 × 4) ÷ (2 × 1) = 20 ÷ 2 = 10. It avoids computing giant factorials and is exact.
Both equal 1. There is exactly one way to choose nothing — the empty selection — so nC0 = 1, and exactly one way to choose everything, so nCn = 1. This also follows from the formula because 0! is defined to be 1.
When the same item can be picked more than once and order still does not matter, the count is C(n + r − 1, r) instead of nCr. Choosing 3 scoops from 5 flavours where repeats are allowed gives C(7, 3) = 35. Switch the calculator to the With repetition mode to compute it.
nCr is also called the binomial coefficient and written as a stacked pair of numbers, n over r. It is the coefficient of the term with r factors in the expansion of (x + y) raised to the nth power — which is why it appears all over Pascal's triangle.
A 6-from-49 lottery has 49C6 = 13,983,816 equally likely combinations, so a single ticket has about a 1 in 14 million chance. Because order does not matter on the ticket, every distinct set of six numbers is one combination — and there are nearly 14 million of them.
Keep going
Related calculators
Keep reading
Further reading
Calculator
What Is the Mean? Definition, Formula, and How to Calculate It
The mean is the average of a set of numbers: add them all up and divide by how many there are. This guide shows the formula, a worked example, and how the mean differs from the median and mode.
Ugo Charles
Statistics
How to Calculate Standard Deviation in Excel
Use STDEV.S or STDEV.P in Excel, choose the right one for your data, and check the result against a worked example.
Ugo Charles
Calculator
How to Calculate Range in Statistics (Beginner’s Guide with Real-Life Examples)
A friendly, beginner-focused guide to understanding and calculating range in statistics. Learn what range means, how to compute it, real-life applications, and common mistakes to avoid.
Ugo Charles
Written by
UC
Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
Did this solve your problem?