C Combination Calculator — C(n, r)

Calculate the number of ways to choose r items from n items when order does not matter. C(n, r) = n! / (r! × (n − r)!). Includes Pascal's triangle, step-by-step working, and symmetry property.

Combinations nCr — Unordered Selections

BrainyCalculators editorial insight — unique to this tool

C(n,r) counts ways to choose r items from n without caring about order — a poker hand is C(52,5) = 2,598,960 possible five-card combos, not permutations because suit order in your hand does not matter. Lottery math ("6 choose 49") and committee selection (pick 3 of 10 board members) are classic nCr problems. The formula n! / (r!(n−r)!) grows fast: C(20,10) = 184,756.

When to use this calculator

Use combinations when order does not matter. When order matters (PIN codes, race podiums, password digit positions), use Permutation instead.

Reference	Value	Context
C(52,5)	2,598,960	Poker hands
C(10,3)	120	Committee picks
C(n,0)	1	Empty selection
C(n,1)	n	Pick one

Not what you need? For ordered arrangements (lock codes, rankings), use Permutation. For event likelihood with known probability, use Probability.

Does order of selection matter?

This page counts unordered groups (n choose r). For ranked arrangements, PINs, and schedules where position matters, use the Permutation Calculator →

n — Total Items

Total number of items to choose from

r — Items to Choose

Number of items to select (order ignored)

Combinations Count Unordered Selections (n Choose r)

A startup hiring committee must pick 3 engineers from 12 applicants — {Ana, Ben, Cara} is the same committee as {Cara, Ana, Ben}. Order of interviews does not change the final team, so we count combinations C(12,3) = 220, not permutations. The formula n! / (r!(n−r)!) removes the r! orderings that describe the same subset.

BrainyCalculators also surfaces Pascal-row symmetry C(n,r) = C(n,n−r) and factorial breakdowns so you can verify by hand. Lottery mathematics is the public face of combinations: UK 6/59 is C(59,6) ≈ 45 million equally likely tickets — your single ticket’s share is the reciprocal of that count.

If arrangement matters — podium finishes, PIN digit positions, seating charts — open Permutation instead. If you are estimating sampling uncertainty in surveys, Sample Size and Confidence Interval belong to a different branch of statistics. Combinations answer “how many groups of size r?” not “how many ordered lists?”

Combination Formula for Unordered Groups

C(n, r) = n! / ( r! × (n − r)! )

Example: C(12, 5) = (12 × 11 × 10 × 9 × 8) / (5 × 4 × 3 × 2 × 1)

Symmetry: C(n, r) = C(n, n − r)

The r! term is the key: it removes every rearrangement of the same selected group. A five-person audit panel is unchanged if the names are listed alphabetically, by seniority, or in the order they accepted the invitation.

Use the symmetry shortcut when r is larger than half of n. Choosing 9 items to include from 12 is the same count as choosing 3 items to leave out, so C(12, 9) = C(12, 3).

Worked Example: Select 5 Reviewers from 12

1

Identify the decision

The final reviewer panel has 5 people from a pool of 12. No chair, rank, or seat is assigned, so order is irrelevant.
2

Apply n choose r

C(12, 5) = 12! / (5! × 7!). Cancel 7! before multiplying to keep the arithmetic small.
3

Multiply only the active run

(12 × 11 × 10 × 9 × 8) / (5 × 4 × 3 × 2 × 1) = 95,040 / 120.
4

Interpret the answer

There are 792 possible panels. The same five names counted in a different order are still one panel, not 120 separate outcomes.

How the Combination Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Formula Used

C(n, r) = n! / (r! * (n - r)!)

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

Enter raw values or summary statistics.
Clean separators and count the sample size.
Apply the relevant statistic, probability, or confidence formula.
Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

Samples should be representative of the population being studied.
Normality or independence assumptions apply only where the selected method requires them.
Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

What is a combination?

A combination is a selection of r items from a set of n items where order does not matter. For example, choosing 3 toppings for a pizza from a list of 10 is a combination problem — the order you pick them makes no difference to the final pizza.

Combination vs permutation?

Use a combination when the group composition is what matters (selecting a committee, choosing lottery numbers). Use a permutation when the order or arrangement matters (assigning first/second/third place, creating a password). P(n,r) = C(n,r) × r!.

What is Pascal's triangle?

Pascal's triangle is a triangular array where each number equals the sum of the two numbers directly above it. Row n gives the binomial coefficients C(n,0), C(n,1), ..., C(n,n). It is used in combinatorics, probability, and algebra (binomial theorem).

Lottery example with combinations?

The UK Lotto draws 6 balls from 59. The number of possible tickets is C(59, 6) = 45,057,474. For the classic 6/49 lottery: C(49, 6) = 13,983,816. Your chance of winning with one ticket is roughly 1 in 14 million.

Real-World Applications

🎟️

Lottery Ticket Universes

A lottery ticket is a set of numbers, not a draw order. For a 6-from-59 game, C(59, 6) counts every unique ticket once; writing the same six numbers in another order does not create a new entry.

🃏

Poker Hand Counting

Five-card poker hands are unordered collections from a 52-card deck. C(52, 5) gives the denominator for comparing flushes, full houses, and rare hands such as royal flushes.

👥

Review Panels and Juries

Selecting 6 jurors from a pool of 40 is a combination problem because the panel has no numbered positions. The count changes only when alternates or foreperson roles are assigned.

🧪

Drug Pair Screening

Testing pairwise interactions among 14 candidates requires C(14, 2) studies. Candidate A with B is the same pair as B with A, so permutations would double-count the lab work.

📦

Bundle and Menu Design

Retailers designing three-item bundles from a catalogue use combinations when shelf order is irrelevant. This helps estimate how many bundles a promotion calendar can offer.

🔐

Access Group Coverage

Security teams can count how many two-person or three-person approval groups exist without implying who approves first. Escalation order, if required, turns the model into a permutation.

Common Mistakes

Using a Combination When Order Matters

Combinations count unordered selections — {A, B, C} = {C, A, B}. If the order of selection matters (first place vs second place, passwords, PIN codes), you need a permutation (P(n,r) = n!/(n-r)!), which is r! times larger than C(n,r).

Forgetting to Check r ≤ n

You cannot choose more items than are available. C(5, 7) is undefined (and equals 0 by convention). Always verify that the number of items chosen (r) does not exceed the total available (n) before applying the formula.

Confusion with Combinations with Repetition

Standard combinations assume no repetition — each item is chosen at most once. If repetition is allowed (e.g., ice cream flavours where you can choose the same flavour twice), the formula is C(n+r-1, r) — also called a multiset coefficient, or "stars and bars" formula.

Computing Large Factorials Directly

Computing 52! directly produces astronomical numbers (~8×10^67). Always simplify by cancelling terms before multiplying: C(52, 5) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1), not 52!/47!/5! in separate steps. This calculator uses BigInt arithmetic to handle large values exactly.

Assuming C(n,r) Always Grows with r

Due to the symmetry C(n,r) = C(n, n-r), combination values increase from r=0 to r=⌊n/2⌋ and then decrease symmetrically. C(10, 8) = C(10, 2) = 45. Students often expect the value to always increase as r increases, which is only true up to the midpoint.

Combination vs Permutation Comparison

Scenario	C(n,r)	P(n,r)	Ratio P/C
Choose 2 from 5	C(5,2) = 10	P(5,2) = 20	2× = 2!
Choose 3 from 10	C(10,3) = 120	P(10,3) = 720	6× = 3!
Choose 4 from 8	C(8,4) = 70	P(8,4) = 1,680	24× = 4!
Choose 5 from 52	C(52,5) = 2,598,960	P(52,5) = 311,875,200	120× = 5!

References

Graham, R. L., Knuth, D. E. & Patashnik, O. Concrete Mathematics, 2nd ed. Addison-Wesley, 1994.
Rosen, K. H. Discrete Mathematics and Its Applications, 8th ed. McGraw-Hill, 2019.
Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 1968.
Stanley, R. P. Enumerative Combinatorics, Vol. 1, 2nd ed. Cambridge University Press, 2011.
Pascal, B. Traité du triangle arithmétique. 1653 (published posthumously 1665).