🔢 Binomial Coefficient Calculator
Calculate binomial coefficient
How to Use This Calculator
Enter n (Total Items)
Type the total number of items (n) in the first field. For example, if you have 10 items, enter 10.
Enter k (Items to Choose)
Type the number of items you want to choose (k). For example, to choose 3 items from 10, enter 3. Note: k must be ≤ n.
Click Calculate
Press the "Calculate" button to compute the binomial coefficient C(n,k).
View Result
See the binomial coefficient value along with the formula breakdown.
Formula
C(n,k) = n! / (k!(n-k)!)
where n! = n × (n-1) × ... × 2 × 1 (factorial)
Example 1: C(5, 2) - Choose 2 items from 5
C(5, 2) = 5! / (2!(5-2)!)
C(5, 2) = 5! / (2! × 3!)
C(5, 2) = (5 × 4 × 3 × 2 × 1) / ((2 × 1) × (3 × 2 × 1))
C(5, 2) = 120 / (2 × 6) = 120 / 12 = 10
There are 10 ways to choose 2 items from 5
Example 2: C(10, 3) - Choose 3 items from 10
C(10, 3) = 10! / (3!(10-3)!)
C(10, 3) = 10! / (3! × 7!)
C(10, 3) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120
There are 120 ways to choose 3 items from 10
Example 3: C(6, 0) - Choose 0 items from 6
C(6, 0) = 6! / (0!(6-0)!)
C(6, 0) = 6! / (1 × 6!) = 1 / 1 = 1
There is exactly 1 way to choose 0 items (do nothing)
About Binomial Coefficient Calculator
The Binomial Coefficient Calculator computes C(n,k), which represents the number of ways to choose k items from a set of n items without regard to order. Also known as "n choose k" or combination, this concept is fundamental in combinatorics, probability, and statistics.
When to Use This Calculator
- Combinatorics: Count combinations and arrangements
- Probability: Calculate probabilities in combination problems
- Statistics: Compute binomial coefficients for binomial distributions
- Pascal's Triangle: Generate entries in Pascal's triangle
- Poker & Games: Calculate possible hands or outcomes
- Computer Science: Solve algorithmic problems involving combinations
Why Use Our Calculator?
- ✅ Instant Results: Calculate large factorials quickly
- ✅ Shows Formula: Displays the step-by-step calculation
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Easy to Use: Simple interface for all skill levels
- ✅ Handles Large Numbers: Efficiently computes large combinations
- ✅ Completely Free: No registration required
Understanding Binomial Coefficients
The binomial coefficient C(n,k) answers the question: "In how many ways can I choose k items from n items?" Order doesn't matter, so the set containing A and B is the same as the set containing B and A.
- C(n, 0) = 1 (there's exactly 1 way to choose nothing)
- C(n, n) = 1 (there's exactly 1 way to choose everything)
- C(n, k) = C(n, n-k) (symmetry property)
- C(n, k) appears in Pascal's Triangle
Real-World Applications
Poker: How many ways can you get a flush? C(13, 5) × 4 = 5,148 (choose 5 cards of same suit from 13, times 4 suits).
Lottery: In a 6/49 lottery, there are C(49, 6) = 13,983,816 possible combinations.
Committees: From 10 people, how many 3-person committees? C(10, 3) = 120 ways.
Tips for Using This Calculator
- n and k must be non-negative integers
- k cannot exceed n (you can't choose more items than you have)
- For large values, the calculator uses efficient computation methods
- C(n,k) = C(n, n-k), so C(10, 7) = C(10, 3)
- Remember: combinations don't consider order; permutations do
Frequently Asked Questions
What's the difference between C(n,k) and P(n,k)?
C(n,k) is combination (order doesn't matter), P(n,k) is permutation (order matters). For example, C(3,2) = 3 (AB, AC, BC) while P(3,2) = 6 (AB, BA, AC, CA, BC, CB).
Can k be greater than n?
No, k must be ≤ n. You cannot choose more items than you have. The calculator will show an error if k > n.
Why is C(n, 0) always 1?
There's exactly one way to choose zero items from any set: choose nothing. This is a fundamental property of combinations.
What about negative numbers?
Binomial coefficients are only defined for non-negative integers. The calculator requires n ≥ 0 and k ≥ 0.
How is this related to Pascal's Triangle?
Each entry in Pascal's Triangle is a binomial coefficient. Row n contains C(n,0), C(n,1), ..., C(n,n).
Can this calculator handle large numbers?
Yes, the calculator can compute large binomial coefficients efficiently. However, very large factorials may take longer to calculate.
Is this calculator accurate for probability calculations?
Yes! Binomial coefficients are essential for calculating probabilities in binomial distributions and combination problems.