Calculate "n choose r" combinations where order does not matter: standard nCr, repetition allowed, and Pascal's triangle binomial coefficients.
Click "Calculate Combinations" to compute options.
A poker player evaluates total possible 5-card hands dealt from a standard 52-card deck: calculating โ
โCโ
= 52! / (5! ร 47!) = 2,598,960 distinct hands. A manager forming a 3-person project committee out of 12 team members calculates total unique group selections: โโCโ = 220. An ice cream shop offering 10 flavors where a customer can choose 3 scoops with repetition allowed evaluates C(10+3-1, 3) = โโCโ = 220 flavor combinations.
A combination is a selection of items from a set where the order of selection does NOT matter. Selecting items {A, B, C} is identical to selecting {C, B, A}.
The standard formula for selecting r items from a total pool of n distinct items without replacement is nCr = n! / (r! ร (n - r)!). In notation, it is often written as (n choose r) or binom(n, r). When repetition is allowed, the formula becomes binom(n + r - 1, r). Binomial coefficients also correspond to the entries of Pascal's Triangle. This tool computes exact combinations using BigInt arithmetic. The following sections explain binomial formulas, symmetry properties (nCr = nC(n-r)), and practical applications in statistics, gaming, and computer science.
When inputs are submitted, the engine uses multiplicative cancels to compute nCr = (n ร (n-1) ร ... ร (n-r+1)) / r! using exact BigInt precision.
1. Standard Combinations Without Repetition (nCr):
nCr = n! / (r! ร (n - r)!) = โ (n - i) / (i + 1) for i = 0 to r - 1.
Symmetry property: nCr = nC(n-r) (e.g. โ
โCโ
= โ
โCโโ).
2. Combinations With Repetition Allowed:
Choosing r items from n categories with unlimited replacement is equivalent to placing r stars between n - 1 bars ("Stars and Bars" theorem):
C_rep(n, r) = binom(n + r - 1, r) = (n + r - 1)! / (r! ร (n - 1)!)
3. Pascal's Triangle & Binomial Theorem:
The expansion of (a + b)โฟ contains coefficients given by row n of Pascal's Triangle:
(a + b)โฟ = โ [binom(n, k) ร aโฟโปแต ร bแต] for k = 0 to n.
Card games and casino odds. Poker, blackjack, and lottery analysis evaluate winning probabilities by dividing favorable card combinations by total nCr sample spaces.
Committee and team selection. Project managers and HR recruiters compute distinct panel combinations when assigning equal-rank team members.
Binomial probability distributions. Statisticians calculate coin flip probabilities and quality control defect rates using the binomial probability formula P(X = k) = binom(n, k) ร pแต ร (1-p)โฟโปแต.
Machine learning feature selection. Data scientists evaluate subsets of r features selected from n candidate variables when training predictive models.
Remember: If order DOES matter (e.g. 1st place, 2nd place, 3rd place rankings), use our Permutation Calculator (nPr) instead.
Use Mode 3 (Pascal's Triangle) to generate all binomial coefficients for expanding algebraic powers (x + y)โฟ.
For single factorials ($n!$), pair this tool with our Factorial Calculator. For basic event probabilities, use our Probability Calculator.
The calculation engine operates client-side in JavaScript using BigInt specifications. Combination โโโโCโ โโ evaluates in under 5 milliseconds.
| Feature | This Tool | Hand Factorials | Scientific Calculator |
|---|---|---|---|
| Max n Limit | n = 1000 (BigInt) | n = 10 practical | n = 99 (Float overflow) |
| Order Matters | No (nCr) | No | nCr key |
| Repetition Allowed | Yes (Stars & Bars) | Manual formula | Not listed |
| Pascal's Triangle | Full row generator | Manual addition | Not listed |
| Privacy | Client-side browser | Paper | Local device |
| Cost | Free | Free | Free |
nCr calculates selections where order does NOT matter ({A,B} = {B,A}). nPr calculates arrangements where order DOES matter (AB โ BA). nCr = nPr / r!.
There is exactly 1 way to choose 0 items from a set of n items: choosing the empty set. Mathematically, nC0 = n! / (0! ร n!) = n! / (1 ร n!) = 1.
There are exactly โ
โCโ
= 2,598,960 possible 5-card poker hands.
Permutation Calculator โ Calculates nPr permutations where order matters.
Probability Calculator โ Computes single, independent, and conditional event probabilities.
Factorial Calculator โ Evaluates exact BigInt factorials and Stirling approximations.