Binomial Coefficient Calculator
Calculate C(n,k) combinations with step-by-step solutions and Pascal's Triangle visualization
C(n, k) = n! / (k! × (n-k)!)
Also known as "n choose k" - the number of ways to choose k items from n items
Must be a non-negative integer
Must satisfy 0 ≤ k ≤ n
Quick Examples
Invalid Input
C(, ) = " choose "
ways to choose items from items
Pascal's Triangle Row
Σ C(, k) for k = 0 to
= 2 (total subsets of a set with elements)
! (n factorial)
! (k factorial)
! ((n-k) factorial)
Step-by-Step Solution
Identify values
n = , k =
Apply the formula
C(, ) = ! / (! × !)
Calculate factorials
= / ( × )
Final result
C(, ) =
Symmetry Property
By symmetry: C(, ) = C(, )
Choosing items
Leaving out items
Pascal's Triangle Row
Position is highlighted (C(, ) = )
Common Binomial Coefficients
| C(n,k) | Value | Application |
|---|---|---|
| C(6, 2) | 15 | Handshakes among 6 people |
| C(10, 3) | 120 | 3-person committees from 10 |
| C(52, 5) | 2,598,960 | 5-card poker hands |
| C(49, 6) | 13,983,816 | Lottery 6/49 combinations |
| C(n, 0) | 1 | One way to choose nothing |
| C(n, n) | 1 | One way to choose all |
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About Binomial Coefficient Calculator
What is the Binomial Coefficient?
The binomial coefficient C(n, k), also written as "n choose k" or (n k), represents the number of ways to choose k distinct items from a set of n distinct items, without regard to the order of selection.
Mathematical Formula
The binomial coefficient is calculated using the formula:
C(n, k) = n! / (k! × (n-k)!)
Where:
- n! (n factorial) = n × (n-1) × (n-2) × ... × 2 × 1
- k is the number of items to choose
- n is the total number of items
Alternative Formulas
Multiplicative Formula
C(n, k) = [n × (n-1) × ... × (n-k+1)] / [k × (k-1) × ... × 1]
Pascal's Identity (Recursive)
C(n, k) = C(n-1, k-1) + C(n-1, k)
Key Properties
Symmetry Property
C(n, k) = C(n, n-k)
Choosing k elements from n is equivalent to choosing (n-k) elements to leave out.
Base Cases
- C(n, 0) = 1 (one way to choose nothing)
- C(n, n) = 1 (one way to choose everything)
- C(n, 1) = n (n ways to choose one item)
Sum Property
For any n: C(n, 0) + C(n, 1) + ... + C(n, n) = 2^n
This represents the total number of subsets of a set with n elements.
Pascal's Triangle
Binomial coefficients form Pascal's Triangle, where each number is the sum of the two numbers directly above it:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Row n contains the values C(n, 0), C(n, 1), ..., C(n, n).
Applications
1. Combinatorics
- Counting combinations in probability problems
- Committee selection problems
- Card game probabilities
2. Probability & Statistics
- Binomial distribution formula
- Lottery probability calculations
- Sampling without replacement
3. Algebra
- Binomial Theorem: (a + b)^n = Σ C(n, k) × a^(n-k) × b^k
- Polynomial expansions
4. Computer Science
- Algorithm complexity analysis
- Bezier curves in graphics
- Combinatorial optimization
Common Examples
| n | k | C(n,k) | Meaning |
|---|---|---|---|
| 5 | 2 | 10 | Ways to choose 2 items from 5 |
| 6 | 3 | 20 | Ways to choose 3 items from 6 |
| 10 | 5 | 252 | Ways to choose 5 items from 10 |
| 52 | 5 | 2,598,960 | Possible 5-card poker hands |
Important Notes
- Constraints: n ≥ k ≥ 0 (both must be non-negative integers, and n must be at least as large as k)
- Result: Always a positive integer
- Symmetry: Use C(n, n-k) when k > n/2 for easier calculation