All calculatorsCombination Calculator
Calculate combinations (choose without order) using C(n, k), also called the binomial coefficient. Get an exact answer (no rounding) with optional step-by-step and a mini Pascal row visual.
Background
A combination counts how many ways to choose k items from n items where order doesn’t matter. The standard notation is C(n,k) (or n \choose k).
How to use this calculator
- Enter n (total items) and k (how many you choose).
- Click Calculate to compute C(n,k).
- Turn on Steps to see the rule and the exact calculation path.
- Use Quick picks for common homework-style cases.
How this calculator works
- Validates integers and checks 0 ≤ k ≤ n.
- Computes C(n,k) using an exact BigInt multiplicative method (no factorial overflow, no rounding).
- Uses symmetry: C(n,k)=C(n,n−k) to reduce work.
Formula & Equation Used
Standard: C(n,k)=\dfrac{n!}{k!(n-k)!}
Product form (used for exact computation): C(n,k)=\prod_{i=1}^{k}\dfrac{n-k+i}{i}
Example Problems & Step-by-Step Solutions
Example 1 — 10 choose 3
- Use product form: C(10,3)=\frac{8}{1}\cdot\frac{9}{2}\cdot\frac{10}{3}
- Compute exactly: 120
Example 2 — 52 choose 5 (poker hand)
How many 5-card hands can you draw from a standard 52-card deck?
- Order doesn’t matter, so use combinations: C(52,5).
- Use product form: C(52,5)=\frac{48}{1}\cdot\frac{49}{2}\cdot\frac{50}{3}\cdot\frac{51}{4}\cdot\frac{52}{5}
- Compute exactly: C(52,5)=2,598,960
Example 3 — 20 choose 2 (pairs)
A class has 20 students. How many different pairs of students can you form?
- Choosing 2 students from 20 where order doesn’t matter: C(20,2).
- Use the shortcut: C(n,2)=\dfrac{n(n-1)}{2}
- Plug in n=20: C(20,2)=\dfrac{20\cdot19}{2}=190
Frequently Asked Questions
Q: What’s the difference between combinations and permutations?
Combinations ignore order. Permutations care about order.
Q: Is C(n,k) the same as a binomial coefficient?
Yes — C(n,k) is the binomial coefficient n \choose k.
Q: Why does this calculator handle big values so well?
It uses exact BigInt arithmetic (no rounding) and avoids factorial blowups.
