The total number of items in your set
How many items you want to select from the set
Enter values to calculate combinations
Combinations calculate how many ways you can choose items from a set where order doesn't matter. Use this for lottery probabilities, committee selections, or any scenario where arrangement is irrelevant.
Our combination calculator is a powerful mathematical tool that helps you calculate the number of ways to choose items from a set where order doesn't matter. Whether you're calculating lottery probabilities, committee selections, team formations, or any scenario involving combinations, this calculator provides instant, accurate results using the fundamental nCr formula.
The calculator supports both combinations with and without repetition, making it versatile for various mathematical problems. Simply enter the total number of items (n) and how many you want to choose (r), and get your combination results instantly with detailed explanations and formula breakdowns.
Using our combination calculator is simple and intuitive:
The calculator automatically updates as you type, providing real-time feedback and ensuring accuracy in your combination calculations.
Combinations use specific mathematical formulas depending on whether repetition is allowed:
Used when you can't select the same item multiple times.
Used when items can be selected multiple times.
Where:
Combination calculations are essential in many practical scenarios:
Calculate odds of winning, number of possible outcomes
Form committees, sports teams, project groups
Calculate gene combinations, trait possibilities
Sample testing, batch selection methods
Guest seating, menu combinations, activity scheduling
Sample group selection, experimental design
A lottery where you choose 6 numbers from 49:
This means you have a 1 in 13,983,816 chance of winning.
Choosing 3 people from a group of 10 for a committee:
There are 120 different ways to form this committee.
Choosing 3 scoops from 5 flavors where you can repeat flavors:
There are 35 different ways to choose your ice cream.
Combinations ignore order (ABC is the same as BAC), while permutations consider order (ABC is different from BAC). Use combinations when arrangement doesn't matter.
Use combinations with repetition when you can select the same item multiple times, like choosing multiple scoops of the same ice cream flavor or picking balls with replacement.
nCr is the mathematical notation for "n choose r" or combinations. It represents the number of ways to choose r items from a set of n items without regard to order.
No, combinations only work with non-negative integers. Both n and r must be whole numbers, and n must be greater than or equal to r (unless using combinations with repetition).
Our calculator uses high-precision mathematical operations and is accurate for all practical purposes. Results are displayed as whole numbers since combinations always yield integer values.
The calculator can handle values up to approximately n=170 due to factorial limitations. For larger values, results would exceed standard number representations.
Combinations are fundamental to probability theory. They help calculate the total number of possible outcomes, which is essential for determining probabilities in events like lotteries, card games, and statistical experiments.
Understanding combinations involves several important mathematical concepts:
Combinations are also known as binomial coefficients and appear in the binomial theorem, probability distributions, and Pascal's triangle.
The values in Pascal's triangle are combination values. Each row contains combinations of different n values, making it useful for quick reference.
Combinations have the property that C(n,r) = C(n,n-r). This means choosing r items from n is the same as leaving out n-r items.