Binomial Coefficient Calculator
Calculate any binomial coefficient C(n, k), also called "n choose k." Enter your values of n and k to get the result with the full factorial expansion shown.
The binomial coefficient C(n, k) counts the number of ways to choose k items from a set of n items, where order does not matter. It shows up everywhere in math, from probability to algebra.
The Formula:
C(n, k) = n! / (k! × (n - k)!)
Where n! (n factorial) means n × (n-1) × (n-2) × ... × 2 × 1.
Quick Example:
How many ways can you pick 3 students from a class of 10?
C(10, 3) = 10! / (3! × 7!) = 720 / 6 = 120 ways.
Key Properties:
- Symmetry: C(n, k) = C(n, n-k). Choosing 3 items from 10 is the same count as choosing 7 items from 10.
- Pascal's Rule: C(n, k) = C(n-1, k-1) + C(n-1, k). Each entry in Pascal's triangle is the sum of the two entries above it.
- Sum of a row: C(n, 0) + C(n, 1) + ... + C(n, n) = 2^n.
Where You See Binomial Coefficients:
- Probability (how many ways an event can happen)
- The Binomial Theorem: (a + b)^n = sum of C(n, k) × a^(n-k) × b^k
- Pascal's Triangle
- Counting problems in combinatorics