Simple combination: C(5, 2)
How many two-person teams from five distinct people?
- Multiply: 5 × 4 = 20.
- Divide: 20 ÷ 2 = 10.
Result: C(5, 2) = 10
Combinatorics notebook
Binomial Coefficient Calculator
Count how many ways you can choose k items from n without order. Type non-negative integers and read C(n, k) as an exact whole number.
Built for probability sets, Pascal rows, and quick homework checks when factorials get unwieldy.
Enter n and k
Use whole numbers only. The tool updates as you type and keeps exact integer arithmetic for results.
Binomial coefficient
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This is C(n, k), the number of unordered selections of size k from n distinct items.
Enter both n and k to see C(n, k).
Using this calculator
- Type n, the size of the full set, then k, how many you select without caring about order.
- Read C(n, k) below. The notation line shows the same pair you entered.
- Tap Clear inputs to start a new problem. Nothing is sent to a server.
What Is a Binomial Coefficient?
A binomial coefficient counts how many ways you can choose k items from n distinct items when order does not matter. Standard notations include C(n, k), (n k), and "n choose k" or "n choose r" when the selected count is written as r.
The coefficient answers a combination question: how many different subsets of size k exist inside a set of size n? It does not count arrangements. ABC and CBA are the same combination of three letters even though they are different permutations.
You use binomial coefficients in combinatorics, discrete mathematics, algebra, probability, statistics, and computer science whenever you model unordered selections, expand binomials, or read coefficients from Pascal's triangle.
Definition in plain language
C(n, k) is the size of the collection of all k-element subsets you can form from an n-element set.
Combination notation
Writers may use (n k), C(n,k), or nCk. In probability texts, r often replaces k: "n choose r."
Real-world applications
Committees, lottery lines, sampling without replacement, and the binomial probability formula all lean on the same counting idea.
Binomial Coefficient Formula
C(n, k) = n! / (k! × (n − k)!)
Same idea with r instead of k:
(n r) = n! / (r! × (n − r)!)
Symmetry property:
C(n, k) = C(n, n − k)
Product form (stable for hand and computer work):
C(n, k) = (n × (n − 1) × … × (n − k + 1)) / (k × (k − 1) × … × 1)
The factorial formula is the definition most textbooks print first. The numerator counts ordered selections if you kept going for k picks; dividing by k! removes reorderings inside the chosen group.
Symmetry matters because C(20, 18) is easier as C(20, 2). Always check whether k is larger than n/2 before you multiply long chains.
Our calculator at the top of this page uses the product form with exact integer division so you do not lose precision on large n.
How to Calculate Binomial Coefficients
Pick the method that matches your tools. All valid routes should agree when 0 ≤ k ≤ n and n, k are integers.
Step 1: Confirm you have a combination problem (order ignored). Step 2: Identify n and k. Step 3: Apply the formula or a shortcut. Step 4: Sanity-check with symmetry or a smaller case you know by heart.
Manual calculation
Write the product of k descending factors from n, then divide by k!. Example: C(8, 3) = (8×7×6)/6 = 56.
Scientific calculator
Many calculators expose nCr or COMB(n,k). Enter n, then k, and read the integer result. Watch for overflow on large n.
Spreadsheet
Excel and Google Sheets use COMBIN(n, k). Build a small table of n and k pairs when you audit homework or lab data.
Shortcut methods
Use symmetry, Pascal's triangle, or recursive identities such as C(n,k) = C(n−1,k−1) + C(n−1,k) when they speed your workflow.
Binomial Coefficient Examples
Each example states n, k, and the count. Scroll to the calculator at the top to verify any pair instantly.
Lottery-style count: C(49, 6)
Classic "pick 6 from 49" without regard to order.
- Note: Use symmetry or a trusted tool; the value is 13,983,816.
Result: C(49, 6) = 13,983,816
Committee selection: C(12, 4)
Choose four reviewers from twelve candidates.
- Product: (12×11×10×9) / (4×3×2×1) = 11,880 / 24.
- Result: 495 committees.
Result: C(12, 4) = 495
Probability setup: C(10, 3)
Count success patterns before attaching probabilities.
- Compute: (10×9×8)/6 = 120 equally likely success subsets if each subset is equally likely.
Result: C(10, 3) = 120
Algebra link: coefficient of x³ in (1 + x)⁷
The coefficient is C(7, 3) by the binomial theorem.
- Apply: C(7, 3) = 35.
Result: 35
n Choose r Calculator
"n choose r" is another name for the binomial coefficient when r denotes how many items you select. On this page, the calculator labels are n (total) and k (chosen); mathematically k and r play the same role.
Permutation problems count ordered lists: P(n, r) = n! / (n − r)!. Combination problems count unordered subsets: C(n, r) = n! / (r! × (n − r)!). If order matters, do not use the binomial coefficient.
C(n, r) = (n r) = n! / (r! (n − r)!)
P(n, r) = n! / (n − r)! (permutations, order matters)
Relationship: P(n, r) = r! × C(n, r)
Binomial Coefficients in Pascal's Triangle
Pascal's triangle lists binomial coefficients by row. Row n (starting at n = 0) contains C(n, 0), C(n, 1), …, C(n, n).
Each interior entry equals the sum of the two entries directly above it. That recursion is the same identity behind row generation in combinatorics courses.
Row pattern
Row 5 reads 1, 5, 10, 10, 5, 1 because those are C(5,0) through C(5,5).
Recursive property
C(n, k) = C(n − 1, k − 1) + C(n − 1, k) for 0 < k < n.
Mathematical interpretation
The triangle encodes how many paths reach each cell if you only move down-left or down-right, linking counting to lattice paths.
Binomial Coefficients in Probability
In a binomial experiment you repeat n independent trials with two outcomes (success/failure). The number of ways to arrange exactly k successes among n trials is C(n, k).
Once you have the count, multiply by p^k (1 − p)^(n − k) for the binomial probability mass function when success probability is p.
Binomial probability
P(X = k) = C(n, k) p^k (1 − p)^(n − k) for k = 0, 1, …, n.
Sampling without replacement
Hypergeometric models replace C(n,k) with adjusted counts when draws change the pool; use the right model for your story.
Event combinations
List outcomes, count favorable subsets with C(n,k), then divide by total equally likely outcomes when the sample space is uniform.
Binomial Theorem and Coefficients
(a + b)^n = Σ C(n, k) a^(n−k) b^k for k = 0 to n
Special case (a + x)^n:
Coefficient of x^k is C(n, k).
Expanding (a + b)^n produces n + 1 terms. Each term picks k copies of b and n − k copies of a; the coefficient counts how many sign patterns produce that split.
Polynomial exercises often ask for a single coefficient without full expansion. Identifying k and reading C(n, k) is faster than multiplying everything out.
Binomial Coefficient Calculator
The tool at the top of this page accepts two non-negative integers: n (total items) and k (chosen items). It returns C(n, k) using exact integer arithmetic.
Results update as you type. Invalid input (decimals, k > n, or values above the safety cap) shows a clear message instead of a misleading number.
Example checks you can run now: C(10, 3) = 120, C(7, 2) = 21, C(0, 0) = 1.
Common Binomial Coefficient Mistakes
Most errors come from using the combination formula on a permutation story, or from factorial overflow on calculators.
Treating order as irrelevant when it matters
Ranking problems need permutations. Seating arrangements are not C(n, k) unless order is explicitly ignored.
Using k > n
You cannot choose more distinct items than exist. The count is zero, not undefined, in standard combinatorics.
Forgetting symmetry
C(100, 98) should be computed as C(100, 2) to save effort.
Mixing with multinomial counts
Splitting objects into three or more labeled groups needs multinomial coefficients, not a single C(n, k).
Rounding large factorials in software
Use integer tools or the product formula. Floating approximations break on moderate n.
Permutations vs Combinations
Both count selections from n, but permutations track order while combinations (binomial coefficients) do not.
| Question | Permutation | Combination (binomial coefficient) |
|---|---|---|
| Order matters? | Yes | No |
| Typical formula | P(n, r) = n! / (n − r)! | C(n, r) = n! / (r! (n − r)!) |
| Example: pick 2 from {A,B,C} | 6 ordered pairs: AB, BA, AC, CA, BC, CB | 3 subsets: {A,B}, {A,C}, {B,C} |
| Relationship | Multiply combination count by r! | Divide permutation count by r! |
If a problem says "arrange" or "rank," start with permutations. If it says "choose," "select," or "form a committee," start with C(n, k).
Deeper guides live in our blog cluster on formulas, examples, and Pascal's triangle; start with the articles linked from the blog index when you want topic-by-topic study paths.
FAQs About Binomial Coefficients
What is the difference between n choose k and n choose r?
They name the same quantity. k and r both stand for how many items you select; n is the pool size.
Why is C(n, 0) equal to 1?
There is exactly one way to choose nothing: the empty subset. Likewise C(n, n) = 1 because you must take all items.
When is C(n, k) zero?
When k > n or when k is negative in combinatorial contexts. There is no valid k-subset of an n-set if k > n.
How does Pascal's triangle relate to C(n, k)?
Entry k of row n equals C(n, k). Each interior cell is the sum of the two cells above it.
Can I use binomial coefficients in the binomial distribution?
Yes. C(n, k) counts success arrangements before you attach probability p to each success pattern.
Does the on-page calculator store my inputs?
No. Calculation runs locally in your browser.
What is the largest n I should enter?
The tool allows n and k up to 1000 for responsiveness. Very large values may take longer but still use exact integers.
How do I check homework by hand?
Use the product formula, symmetry, or Pascal's triangle for small n, then confirm with the calculator.