How to Use Combination Calculator
The Combination Calculator gives you the exact number of combinations in three simple steps.
- Enter n (Total Items) — Provide the size of the full set you are choosing from (e.g. 52 for a standard deck of cards). Must be a non-negative integer ≤ 170.
- Enter r (Items to Choose) — Provide how many items you want to select (e.g. 5 for a poker hand). Must satisfy 0 ≤ r ≤ n.
- Read the Result — The Combination Calculator displays C(n, r), the substituted formula, and a plain-language interpretation.
The Combination Calculator updates instantly — no button press required.
Formula & Theory — Combination Calculator
The Combination Calculator is built on the binomial coefficient formula from combinatorics:
C(n, r) = n! / (r! × (n − r)!)
| Symbol | Meaning |
|---|---|
| n | Total number of distinct items in the set |
| r | Number of items to select |
| n! | Factorial of n: the product 1 × 2 × 3 × … × n |
| C(n, r) | Number of ways to choose r items from n without regard to order |
Why Factorial Division Works
The numerator n! counts all ordered arrangements of n items. Dividing by r! removes the internal orderings within the chosen group, and dividing by (n−r)! removes the orderings of the unchosen items. What remains is the count of distinct, unordered subsets — exactly what the Combination Calculator returns.
Special Cases
- C(n, 0) = 1 for any n (there is exactly one way to choose nothing)
- C(n, n) = 1 for any n (there is exactly one way to choose everything)
- C(n, 1) = n (choosing one item at a time gives n distinct options)
Use Cases for Combination Calculator
The Combination Calculator is an essential tool across mathematics, statistics, and everyday decision-making:
- Lottery & Gambling — Calculate the exact odds of winning a lottery by finding how many ticket combinations exist (e.g. C(49, 6) for a typical 6/49 lottery).
- Card Games — Determine how many distinct 5-card poker hands can be dealt from a 52-card deck using C(52, 5).
- Team Selection — Find how many ways a coach can pick a starting lineup of r players from a squad of n.
- Probability Problems — Compute sample space sizes for binomial probability experiments.
- Statistics & Sampling — Count the number of possible samples of size r from a population of size n.
- Exam & Homework — Quickly verify combination calculations in discrete mathematics or probability courses.
Whenever you need to count unordered selections without repetition, the Combination Calculator delivers the answer instantly and accurately.