Combinations Without Repetition
Calculate the number of combinations (nCr) without repetition using our free online calculator. Enter n total objects and r choices to get instant results with formula breakdown and charts.
About This Calculator
Our Combinations Without Repetition Calculator (also called nCr calculator or binomial coefficient calculator) computes the number of ways to choose r items from a set of n distinct items where order does not matter and each item can be selected only once. This is a fundamental concept in combinatorics and probability theory used by students, teachers, data scientists, and professionals working with probability, statistics, and combinatorial optimization problems.
The calculator uses the standard combinations formula C(n,r) = n! / (r! × (n-r)!), also known as the binomial coefficient. It computes the result using an optimized iterative algorithm that minimizes overflow risk by cancelling common factors in the numerator and denominator. The result represents the total number of distinct subsets of size r that can be formed from a set of n elements. For example, choosing 3 toppings from a menu of 10 offers C(10,3) = 120 different combinations.
Regional Notes
India: Combinations without repetition are widely taught in CBSE and state board mathematics curricula under permutations and combinations. They are used in JEE and other competitive exam problems, lottery calculations (e.g., Kerala Lottery), and cricket team selection scenarios.
United States: Combinations appear in SAT/ACT math sections, AP Statistics coursework, and real-world applications like Powerball lottery odds (C(69,5) for white balls) and poker hand probabilities. They are also used in quality control sampling and experimental design.
United Kingdom: Combinations are part of GCSE and A-Level mathematics, particularly in statistics and probability modules. They are used in National Lottery odds calculations (C(59,6)), sports betting combinations, and academic research study design.
Frequently Asked Questions
What is a combination without repetition?
A combination without repetition (also called nCr or binomial coefficient) is a selection of r items from a set of n distinct items where the order of selection does not matter and no item can be chosen more than once. For example, choosing 3 fruits from a basket of 5 different fruits is a combination without repetition, where {apple, banana, cherry} is the same as {cherry, banana, apple}.
How do you calculate combinations without repetition?
The formula for combinations without repetition is C(n,r) = n! / (r! × (n-r)!), where n is the total number of items, r is the number of items to choose, and ! denotes factorial. For example, C(5,3) = 5! / (3! × 2!) = 120 / (6 × 2) = 10. This is also called the binomial coefficient and is read as n choose r.
What is the difference between combinations and permutations?
The key difference is that combinations do not consider order while permutations do. For combinations, selecting items A, B, C is the same as B, C, A. For permutations, each different ordering counts as a separate arrangement. For example, from 3 items choosing 2, there are C(3,2) = 3 combinations (AB, AC, BC) but P(3,2) = 6 permutations (AB, BA, AC, CA, BC, CB).
What is the formula for combinations with repetition vs without repetition?
Without repetition (standard nCr): C(n,r) = n! / (r! × (n-r)!). With repetition (multichoose): C'(n,r) = C(n+r-1,r) = (n+r-1)! / (r! × (n-1)!). The with-repetition formula allows each item to be selected multiple times, resulting in more possible combinations for the same n and r.
What are real-world applications of combinations without repetition?
Combinations without repetition appear in lottery probability calculations, poker hand evaluations, committee selection problems, quality control sampling, and sports tournament brackets. In India, combinations are used in lottery systems like Kerala Lottery and in competitive exam selection problems. In the US, combinations are used in Powerball and Mega Millions odds calculations. In the UK, they appear in National Lottery odds and cricket team selection.
How many combinations of 5 items taken 3 at a time exist?
There are C(5,3) = 10 combinations of 5 items taken 3 at a time without repetition. The formula is C(5,3) = 5! / (3! × 2!) = 120 / (6 × 2) = 10.
What is the binomial coefficient and how is it related to combinations?
The binomial coefficient, denoted as C(n,k) or (n choose k), is the same as the number of combinations without repetition. It appears in the binomial theorem expansion of (x + y)^n and in probability theory for calculating binomial probabilities. It is also used in Pascal's triangle where each number is the sum of the two numbers above it.
Can n be smaller than r in combinations?
No, in combinations without repetition, n must be greater than or equal to r. If r > n, it is impossible to choose r distinct items from a set of n items since each item can only be selected once. The calculator will return 0 for such cases.