Binomial Coefficient
Calculate n choose k (binomial coefficient nCr) instantly with this free calculator. Supports non-negative integers for combinatorics, probability, and stats.
About This Calculator
The binomial coefficient, written as C(n,k) or nCr, represents the number of ways to choose k items from a set of n distinct items where order does not matter. It is a fundamental concept in combinatorics, probability theory, and statistics.
It is calculated using the formula: C(n,k) = n! / (k! x (n-k)!). For example, C(5,2) = 10 because there are 10 ways to choose 2 items from a group of 5.
Binomial coefficients appear in the binomial theorem (expanding powers of binomials), Pascal's triangle, combinations and permutations, and probability distributions such as the binomial distribution. Whether you're a student learning combinatorics, a data scientist computing probabilities, or a researcher working with statistical models, this calculator helps you compute nCr quickly and accurately.
Frequently Asked Questions
What is the binomial coefficient C(n,k) used for?
The binomial coefficient counts the number of ways to choose k items from n items without regard to order. It is used extensively in probability (binomial probability formula), combinatorics (combinations), algebra (binomial theorem expansion), and statistics (counting outcomes).
How is the binomial coefficient calculated?
The formula is C(n,k) = n! / (k! x (n-k)!), where n! (n factorial) is the product of all positive integers from 1 to n. For instance, C(5,2) = 5! / (2! x 3!) = 120 / (2 x 6) = 10. Our calculator handles all the factorial arithmetic instantly.
What is the difference between combinations and permutations?
Combinations (binomial coefficients) count selections where order does not matter -- choosing 3 team members from 10. Permutations count arrangements where order matters -- arranging 3 books on a shelf. The number of permutations P(n,k) = n! / (n-k)!, which is always larger than C(n,k) for the same n and k.
What is the relationship between binomial coefficients and Pascal's triangle?
Pascal's triangle is a triangular array where each number is the sum of the two numbers above it. The rows of Pascal's triangle correspond to binomial coefficients: row n contains C(n,0), C(n,1), ..., C(n,n). This provides a simple way to compute binomial coefficients for small values without factorials.
What is the range of values for n and r?
Both n and r must be non-negative integers, and r must be less than or equal to n. The maximum supported value for n is 170 to avoid numerical overflow when computing factorials. For values of n up to about 170, the calculator uses standard arithmetic. For larger values, specialized mathematical software is recommended.
What are some real-world applications of nCr?
Binomial coefficients appear in lottery probability calculations (chances of winning), quality control (sampling defective items), genetics (inheritance patterns), machine learning (feature selection), poker hand probabilities, survey sampling, binomial distribution calculations, and even in estimating the number of possible subnetworks in network analysis.
How do I use the binomial coefficient in probability?
In the binomial distribution, the probability of getting exactly k successes in n independent trials is: P(X=k) = C(n,k) x p^k x (1-p)^(n-k), where p is the probability of success on a single trial. The binomial coefficient counts how many sequences of trials produce exactly k successes.