Combinations and Permutations Calculator
Free Combinations permutations Calculator for sequences. Enter values to get step-by-step solutions with formulas and graphs.
Calculator
Adjust values & calculateAll Four Variants
Formula
P(n,r) counts ordered arrangements of r items from n distinct items. C(n,r) counts unordered selections. With repetition: P = n^r and C = (n+r-1)!/(r!(n-1)!). The factorial n! equals the product of all integers from 1 to n.
Last reviewed: December 2025
Worked Examples
Example 1: Lottery Probability
Example 2: Password Possibilities
Background & Theory
The Combinations and Permutations Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Combinations and Permutations Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
P(n,r) = n!/(n-r)! | C(n,r) = n!/(r!(n-r)!)
P(n,r) counts ordered arrangements of r items from n distinct items. C(n,r) counts unordered selections. With repetition: P = n^r and C = (n+r-1)!/(r!(n-1)!). The factorial n! equals the product of all integers from 1 to n.
Worked Examples
Example 1: Lottery Probability
Problem: A lottery draws 6 numbers from 49 balls (without repetition, order does not matter). How many possible outcomes exist?
Solution: This is a combination without repetition: C(49, 6)\nC(49,6) = 49! / (6! x 43!)\n= (49 x 48 x 47 x 46 x 45 x 44) / (6 x 5 x 4 x 3 x 2 x 1)\n= 10,068,347,520 / 720\n= 13,983,816\nProbability of winning = 1 / 13,983,816
Result: 13,983,816 possible combinations | Winning probability: 1 in ~14 million
Example 2: Password Possibilities
Problem: How many 4-character passwords can be made from 26 lowercase letters? Calculate both with and without repetition.
Solution: With repetition (letters can repeat):\nP = 26^4 = 456,976 passwords\n\nWithout repetition (all letters unique):\nP(26,4) = 26! / 22! = 26 x 25 x 24 x 23 = 358,800\n\nDifference = 456,976 - 358,800 = 98,176\nRepetition adds 27.4% more possibilities
Result: With repetition: 456,976 | Without: 358,800 | 27.4% more with repetition
Frequently Asked Questions
What is the difference between combinations and permutations?
The fundamental difference is whether order matters. Permutations count arrangements where order is important: choosing president, vice president, and treasurer from 10 people is a permutation problem because the same 3 people in different roles count as different outcomes. Combinations count selections where order does not matter: choosing 3 committee members from 10 people is a combination problem because the same 3 people form only one committee regardless of selection order. The permutation count is always greater than or equal to the combination count for the same n and r values because permutations count each unordered group multiple times.
What are the formulas for permutations and combinations?
Permutations without repetition: P(n,r) = n! / (n-r)!, which counts ordered arrangements of r items from n distinct items. Combinations without repetition: C(n,r) = n! / (r!(n-r)!), which counts unordered selections. With repetition allowed, permutations become n^r (each of the r positions can be any of the n items). Combinations with repetition use the multiset formula: C(n+r-1, r) = (n+r-1)! / (r!(n-1)!). The exclamation mark denotes factorial, where n! equals n times (n-1) times (n-2) down to 1. These formulas are cornerstone tools in combinatorics and probability theory.
When should I use permutations versus combinations in word problems?
Use permutations when the problem involves arrangement, ordering, ranking, or assigning distinct roles. Key phrases include: arrange, order, rank, first/second/third, sequence, password, PIN, license plate, and schedule. Use combinations when the problem involves selection, grouping, or choosing without regard to order. Key phrases include: choose, select, committee, team, group, hand of cards, and subset. A helpful test: if swapping two selected items creates a different outcome, use permutations. If swapping does not change anything, use combinations. For example, a 4-digit PIN uses permutations (1234 differs from 4321), while choosing 4 pizza toppings uses combinations.
What does repetition mean in permutations and combinations?
Repetition (also called replacement) means items can be selected more than once. Without repetition, once an item is chosen, it cannot be chosen again, like dealing cards from a deck. With repetition, items remain available for subsequent selections, like rolling a die multiple times where each roll can produce any number. A 4-digit PIN with repetition has 10^4 = 10,000 possibilities (each digit can be 0-9 regardless of other digits). Without repetition, the same PIN has 10 x 9 x 8 x 7 = 5,040 possibilities. Real-world scenarios with repetition include passwords, dice rolls, and sampling with replacement. Without repetition includes lottery draws and card dealing.
How are combinations used in probability calculations?
Combinations form the foundation of many probability calculations. The probability of an event equals favorable outcomes divided by total outcomes, and both are often calculated using combinations. For a lottery drawing 6 numbers from 49, the total outcomes are C(49,6) = 13,983,816. The probability of matching all 6 is 1/13,983,816. In card games, the probability of being dealt a flush in poker requires calculating C(13,5) x 4 favorable hands divided by C(52,5) total hands. The binomial probability formula uses combinations directly: P(X=k) = C(n,k) x p^k x (1-p)^(n-k), which calculates the probability of exactly k successes in n trials.
What is the relationship between combinations and Pascals triangle?
Each entry in Pascals triangle equals a specific combination value. Row n, position r gives C(n,r). Row 0 is just 1. Row 1 is 1, 1. Row 2 is 1, 2, 1. Row 3 is 1, 3, 3, 1. Row 4 is 1, 4, 6, 4, 1. Each number is the sum of the two numbers above it, which corresponds to the identity C(n,r) = C(n-1,r-1) + C(n-1,r). This identity has a beautiful combinatorial interpretation: to choose r items from n, either include the nth item (choosing r-1 more from n-1) or exclude it (choosing all r from n-1). Pascals triangle also reveals binomial coefficients used to expand expressions like (a+b)^n.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy