Financial term · Reviewed August 17, 2026
Combination
Combination: A selection of items where order does not matter. Choosing 3 people for a committee from a group is a combination problem. The Permutation & Combination Calculator returns this figure from ₙpᵣ = n! ÷ (n−r)!; ₙcᵣ = n! ÷ (r!(n−r)!). At that calculator’s documented worked-example inputs the permutations (order matters) is 720; holding every other input steady, moving n (total items) from 8 to 12 moves it from 336 to 1,320, a spread of 984.
Order does not matter in a combination. Pick three people for a committee and the same three are the same committee no matter what sequence you named them in. That single question, does rearranging the selection change the answer, decides whether you are counting combinations or permutations, and getting it wrong does not produce a slightly off answer. It produces one that is wrong by the number of ways the chosen items can be shuffled.
| Term | Combination |
|---|---|
| Category | Statistics & Probability |
| How it is found | ₙPᵣ = n! ÷ (n−r)!; ₙCᵣ = n! ÷ (r!(n−r)!) |
| Worked example | Permutations (order matters): 720 |
| Computed by | Permutation & Combination Calculator |
| Related terms | 5 defined in this glossary |
Before you rely on it
What to check
Ask whether swapping two chosen items produces a different outcome. If not, it is a combination.
The common error
Where people go wrong with combination
Reaching for the permutation formula because it appeared first in the textbook. Committee, hand of cards, pizza toppings, lottery ticket: all combinations, and all routinely counted the other way.
Worked through
Combination in plain numbers
How many ways can you choose 3 people from a group of 10 for a committee? Order does not matter, so use the combination formula: ten factorial divided by three factorial times seven factorial, which is 720 ÷ 6, or 120; ask a different question: how many ways to hand out gold, silver and bronze among the same 10? Now order matters, so it is a permutation: 10 × 9 × 8 = 720, and the permutation count is 6 times larger, since three factorial is 6, because each committee of three can be arranged six ways. A lottery that draws 6 numbers from 49 has 13,983,816 combinations, and only one wins.
Easily confused
Combination and its neighbours
A combination ignores order and a permutation respects it. That is the whole difference, and it explains why the permutation count is always larger for the same n and r. Choosing a committee is a combination. Assigning a president, a treasurer and a secretary is a permutation. Combinations with repetition allow the same item more than once, like choosing scoops of ice cream from flavors. And a combination is not a probability. It is a count, which you divide into another count to get one.
In real decisions
Where combination turns up
Combinations appear in probability, statistics, games and planning. Poker odds come from counting hands. Quality inspectors count possible samples. Researchers count possible groupings before a randomized study. Programmers count test cases; the binomial distribution is built from them, and Pascal's triangle lists them. Whenever a question asks how many ways to pick without caring about order, the answer is a combination. Knowing the difference saves you from reporting answers that are off by a factor of six or more.
Know the edges
Where combination stops being reliable
The numbers grow very fast. Choosing 10 from 100 already exceeds 17 trillion. Calculators may overflow or round for large values, and the exact factorial route is impractical, so they use shortcuts; the formula assumes distinct items and equal chances. If items are identical, or some choices are more likely than others, the count needs adjusting; a count of possibilities also says nothing about which ones you will actually see.
Quick answers
What people ask about combination
How do I write a combination? As nCr or C(n, r), or with the notation n over r. It is read n choose r. For 10 choose 3 the value is 120.
Why divide by r factorial? Because the same group of r items can be listed in r factorial orders. Dividing removes the repeats, so each group counts once.
Put it to work
How combination is calculated
At the permutation & combination calculator’s worked-example inputs, permutations (order matters) is 720, combinations (order doesn't matter) is 120. Moving n (total items) from 8 to 12 moves that first figure from 336 to 1,320.
Open the Permutation & Combination Calculator →
These are standard formulas from introductory statistics. Real-world inference depends on assumptions, normality, independence, a genuinely random sample, that this calculator does not check for you.
Related vocabulary
Terms used alongside this one
Keep reading
Nearby in the glossary
- ← Collateral
- Compound annual growth rate (CAGR) →
- All 174 defined termsThe complete Free Calculators Online financial glossary.