Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Advanced Math Operations > Combination Calculator

Combination Calculator

Calculate combinations C(n,k) with step-by-step solutions, Pascal's Triangle visualization, interactive diagrams, and detailed formula breakdowns for combinatorics problems.

Combination Calculator

Combination Formula

C(n, k) = n! / (k! × (n-k)!)
Quick Presets
C(10,3) C(52,5) Poker C(49,6) Lottery C(20,2) Handshakes C(8,4) C(100,3)

Embed Combination Calculator Widget

About Combination Calculator

Welcome to the Combination Calculator, a comprehensive tool for calculating combinations C(n,k) with step-by-step solutions, Pascal's Triangle visualization, and interactive diagrams. Whether you are solving probability problems, studying combinatorics, calculating lottery odds, or working on counting problems, this calculator provides detailed explanations and visual representations to help you understand the mathematics behind combinations.

What is a Combination?

A combination is a selection of items from a larger set where the order of selection does not matter. It answers the question: "In how many ways can I choose k items from n items?"

For example, if you want to choose 3 students from a class of 10 to form a committee, the combination C(10,3) = 120 tells you there are 120 different possible committees. The order in which you select the students doesn't matter - selecting Alice, Bob, then Carol gives the same committee as selecting Carol, Alice, then Bob.

The Combination Formula

Combination Formula
$$C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Where:

Combinations vs Permutations

The key difference between combinations and permutations is whether order matters:

Aspect Combination Permutation
Order Does NOT matter DOES matter
Example {A, B, C} = {C, B, A} ABC ≠ CBA
Formula n! / (k!(n-k)!) n! / (n-k)!
Use Case Selecting committee members Arranging race finishers

For the same n and k values, permutations always yield larger results because they count each group multiple times (once for each possible ordering).

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The triangle provides a visual way to find combination values:

For example, row 5 of Pascal's Triangle shows: 1, 5, 10, 10, 5, 1, corresponding to C(5,0), C(5,1), C(5,2), C(5,3), C(5,4), C(5,5).

How to Use This Calculator

  1. Enter n (total items): Input the total number of items in your set. The maximum value is 170.
  2. Enter k (items to choose): Input how many items you want to select. This must be less than or equal to n.
  3. Click Calculate: The calculator will compute C(n,k) and display:
    • The final result with thousands separators for readability
    • Step-by-step calculation breakdown
    • Pascal's Triangle visualization (for n ≤ 12)
    • All possible combinations listed (for small results)
    • Real-world application examples
  4. Try preset values: Use the quick preset buttons to explore common combination problems.

Real-World Applications

Lottery and Gambling

Combinations are essential for calculating lottery odds. For a 6/49 lottery (choosing 6 numbers from 49), C(49,6) = 13,983,816 possible combinations, giving odds of about 1 in 14 million.

Probability and Statistics

The binomial probability formula uses combinations: P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where p is the probability of success in a single trial.

Team Selection

When selecting a committee of 5 from 20 candidates, C(20,5) = 15,504 possible committees can be formed.

Card Games

Poker hand probabilities rely on combinations. A standard deck has C(52,5) = 2,598,960 possible 5-card hands.

Handshake Problems

If n people shake hands with everyone exactly once, the total number of handshakes is C(n,2) = n(n-1)/2.

Important Properties of Combinations

Symmetry Property

Symmetry
$$C(n, k) = C(n, n-k)$$

Choosing k items to include is equivalent to choosing (n-k) items to exclude.

Pascal's Identity

Pascal's Identity
$$C(n, k) = C(n-1, k-1) + C(n-1, k)$$

This recursive relationship is why Pascal's Triangle works - each number is the sum of the two above it.

Sum of Row

Row Sum
$$\sum_{k=0}^{n} C(n, k) = 2^n$$

The sum of all combinations in row n equals 2^n, representing all possible subsets of an n-element set.

Frequently Asked Questions

What is a combination in mathematics?

A combination is a selection of items from a larger set where the order of selection does not matter. It is denoted as C(n,k) or "n choose k", representing the number of ways to choose k items from n items. Unlike permutations, combinations treat {A,B,C} and {C,B,A} as the same selection.

What is the formula for combinations?

The combination formula is C(n,k) = n! / (k! × (n-k)!), where n is the total number of items, k is the number of items to choose, and ! denotes factorial. This formula calculates how many different groups of k items can be selected from n items without considering order.

What is the difference between combinations and permutations?

The key difference is order: in combinations, order does not matter (selecting A,B,C is the same as C,B,A), while in permutations, order matters (ABC and CBA are different arrangements). Combinations count groups, permutations count arrangements.

What is Pascal's Triangle and how does it relate to combinations?

Pascal's Triangle is a triangular array where each number is the sum of the two numbers directly above it. The nth row contains the values C(n,0), C(n,1), ..., C(n,n). This provides a visual way to find combination values without calculation.

What are real-world applications of combinations?

Combinations have many practical applications: calculating lottery odds, counting handshakes at a party, determining poker hand probabilities, selecting team members from a group, and solving problems in probability, statistics, and computer science.

Related Calculators

Reference this content, page, or tool as:

"Combination Calculator" at https://MiniWebtool.com/combination-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 18, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Binomial Coefficient Calculator
Binomial Probability Distribution Calculator
Factorial Calculator
Permutation Calculator

Advanced Math Operations:

Top & Updated:

Random PickerRandom Name PickerRelative Standard Deviation CalculatorFPS ConverterMAC Address GeneratorSort NumbersBitwise CalculatorFeet and Inches to Cm ConverterMAC Address LookupLine CounterRemove SpacesWord to Phone Number ConverterBatting Average CalculatorRandom Quote GeneratorPercent Off CalculatorSum CalculatorFacebook User ID LookupNumber of Digits CalculatorLog Base 10 CalculatorAI ParaphraserRoman Numerals ConverterSHA256 Hash GeneratorDecimal to BCD ConverterSun, Moon & Rising Sign Calculator 🌞🌙✨Binary to Gray Code ConverterPhone Number ExtractorERA CalculatorMP3 LooperBCD to Decimal ConverterVideo to Image ExtractorSalary Conversion CalculatorQuotient and Remainder CalculatorCm to Feet and Inches ConverterSaturn Return CalculatorSquare Root (√) CalculatorOn Base Percentage CalculatorRemove Audio from VideoCompound Growth CalculatorFirst n Digits of PiOutlier CalculatorRemove AccentGray Code to Binary ConverterNumber to Word ConverterAI Punctuation AdderDay of Year CalendarTime Duration CalculatorDay of the Year Calculator - What Day of the Year Is It Today?Slugging Percentage CalculatorMerge VideosBinary to BCD ConverterAudio SplitterName Number CalculatorCompare Two StringsWAR CalculatorScientific Notation to Decimal ConverterIP Address to Hex ConverterModulo CalculatorList of Prime NumbersMaster Number CalculatorLove Compatibility CalculatorRandom Birthday GeneratorConvolution CalculatorStandard Error CalculatorPercent Growth Rate CalculatorAverage Deviation CalculatorPercentage Increase CalculatorSum of Positive Integers CalculatorRandom Truth or Dare GeneratorRandom Group GeneratorText FormatterRemove Lines Containing...Ratio to Percentage CalculatorLeap Years ListSHA512 Hash GeneratorPER CalculatorRandom Object GeneratorAntilog CalculatorVideo SplitterOne Rep Max (1RM) CalculatorCube Numbers ListMaze GeneratorMedian CalculatorBCD to Binary ConverterFirst n Digits of eMAC Address AnalyzerProportion Calculator📅 Date CalculatorOPS CalculatorAdd or Replace Audio in VideoURL ExtractorNatural Log CalculatorReverse VideoBonus CalculatorLoop VideoPVIF CalculatorLog Base 2 CalculatorNumber ExtractorHypotenuse CalculatorBreak Line by CharactersHex to BCD ConverterOctal CalculatorEmail ExtractorAdd Prefix and Suffix to TextWHIP CalculatorVideo CompressorSort Lines AlphabeticallyLottery Number GeneratorBCD to Hex ConverterList RandomizerAverage CalculatorPVIFA CalculatorActual Cash Value CalculatorAngel Number CalculatorRoot Mean Square CalculatorSHA3-256 Hash GeneratorAI Sentence ExpanderLbs to Kg ConverterHex to Decimal ConverterRandom String GeneratorRemove Leading Trailing SpacesAmortization CalculatorMarkup CalculatorDecimal to Hex ConverterList of Fibonacci NumbersRandom Number PickerBlood Donation Time CalculatorField Goal Percentage CalculatorRandom Line PickerAPY to APR CalculatorSort Text By LengthName RandomizerRandom Animal GeneratorSquare Numbers ListFixed Deposit CalculatorPPM to Percent ConverterCaesar Cipher ToolMorse Code GeneratorPercent to PPM ConverterSRT to TXT Converterkg to lbs ConverterXML ValidatorArithmetic Mean CalculatorBoard Foot Calculator 🪵Squareness Checker (3-4-5 Rule) 📐Miter Angle Calculator 📐Slope and Grade Calculator 📐Sine CalculatorCosine CalculatorHigh-Precision Tangent CalculatorArcsin CalculatorArccos (Inverse Cosine) CalculatorArctan CalculatorArctan2 CalculatorCosecant/Secant/Cotangent CalculatorHyperbolic Functions CalculatorRight Triangle CalculatorGeneral Triangle SolverLaw of Sines CalculatorLaw of Cosines CalculatorDMS to Decimal Degrees ConverterDecimal Degrees to DMS ConverterPolar to Cartesian ConverterHigh-Precision Cartesian to Polar Coordinates ConverterInteractive Unit Circle VisualizerTrigonometric Identities CalculatorTrigonometric Function GrapherAlgebraic Expression SimplifierRadical SimplifierRational Expression CalculatorPolynomial Factoring CalculatorExpand Polynomials CalculatorPolynomial Long Division CalculatorSynthetic Division CalculatorLinear Equation SolverAbsolute Value Equation SolverRadical Equation SolverSystem of Linear Equations SolverInequality SolverAbsolute Value Inequality SolverSystem of Inequalities GrapherFunction GrapherDomain and Range CalculatorInverse Function CalculatorFunction Composition CalculatorVertex and Axis of Symmetry CalculatorX and Y Intercept CalculatorHTML Entity Encoder/DecoderURL Slug GeneratorJSON String Escape/UnescapeMarkdown Table GeneratorSmart Quotes RemoverDaily Time Savings CalculatorLorem Ipsum GeneratorReadability Score CalculatorKeyword Density CheckerBionic Reading ConverterText to Speech ReaderFancy Text GeneratorZalgo Text GeneratorUpside Down Text GeneratorInvisible Text GeneratorASCII Art GeneratorList Difference CalculatorText Column ExtractorText to SQL List ConverterInvisible Character RemoverStock Average CalculatorPosition Size CalculatorMargin Call CalculatorShort Selling Profit CalculatorRisk of Ruin CalculatorBlack-Scholes Option Pricing CalculatorOption Greeks CalculatorImplied Volatility CalculatorOptions Profit CalculatorImpermanent Loss CalculatorCrypto Arbitrage CalculatorSatoshi to USD ConverterCrypto Leverage CalculatorPivot Point CalculatorFibonacci Extension CalculatorStop Loss & Take Profit CalculatorKelly Criterion CalculatorMartingale Strategy CalculatorCompound Daily Interest CalculatorPip Value CalculatorGaussian Distribution GeneratorRandom Tournament Bracket GeneratorRandom Meal GeneratorRandom Emoji GeneratorRandom Color Palette GeneratorRandom Country GeneratorRandom JSON GeneratorRandom Credit Card GeneratorRandom User-Agent GeneratorRandom Coordinate GeneratorRandom Date GeneratorRandom Time GeneratorRandom IP Address Generator