Combinations Calculator
Calculate the number of ways to choose items from a set without regard to order (nCr).
Combination Results
Number of Combinations (nCr):
Formula:
Calculation:
About Our Combinations Calculator
Our Combinations Calculator is a powerful tool for calculating the number of ways to select items from a set without considering the order. Combinations are fundamental in probability theory, statistics, and many real-world applications including lotteries, team selection, and sampling.
What Are Combinations?
In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter. For example, when selecting a team of 3 people from a group of 10, it doesn't matter whether you select person A, then B, then C, or C, then A, then B - it's considered the same combination.
The Combination Formula
The formula for calculating combinations is:
nCr = n! / (r! × (n-r)!)
Where:
- n is the total number of items in the set
- r is the number of items being chosen
- ! represents the factorial operation (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)
Key Features:
- Calculate combinations (nCr) for any valid values of n and r
- See the step-by-step calculation with the formula breakdown
- Handle large numbers with precision
- User-friendly interface for quick probability calculations
How to Use:
- Enter the total number of items (n)
- Enter the number of items to choose (r)
- Click "Calculate Combinations" to see the results
Real-World Applications:
Lottery Calculations: Find the odds of winning by calculating the number of possible combinations.
Team Selection: Calculate how many different teams can be formed from a group of people.
Card Games: Determine the probability of being dealt certain hands in games like poker.
Statistical Sampling: Calculate the number of possible sample groups in research and surveys.
Committee Formation: Find how many different committees can be formed from a group of candidates.
Combinations vs. Permutations
It's important to understand the difference between combinations and permutations:
- Combinations: The order doesn't matter. Example: Selecting 3 toppings for a pizza from 8 options.
- Permutations: The order does matter. Example: The possible arrangements of 1st, 2nd, and 3rd place in a race with 8 runners.
Our calculator specifically handles combinations (nCr) where order doesn't matter.
Perfect for students, teachers, statisticians, researchers, or anyone working with probability and combinatorial problems. Start calculating combinations today!
Frequently Asked Questions
What's the difference between combinations (nCr) and permutations (nPr)?
The key difference is whether order matters. Combinations (nCr) represent the number of ways to select items without regard to order. Permutations (nPr) represent the number of ways to select AND arrange items where order matters. For example, when forming a committee, the specific order of members doesn't matter (combination), but when ranking contestants in a competition, the order is important (permutation).
Is there a limit to the numbers I can calculate?
Our calculator can handle reasonably large inputs, but factorial calculations grow extremely large very quickly. For practical purposes, calculations with n values above 170 may exceed JavaScript's number precision limits. For most real-world applications, however, the calculator will work perfectly. If you need to calculate extremely large combinations, consider using specialized mathematical software.
Can I calculate combinations where r is greater than n?
No, mathematically speaking, you cannot select more items (r) than exist in the total set (n). If r > n, the number of combinations is zero, and our calculator will indicate this. The valid range for combinations is 0 ≤ r ≤ n.
How can I use combinations to calculate probability?
To calculate the probability of a specific outcome, you can divide the number of favorable combinations by the total number of possible combinations. For example, to find the probability of drawing exactly 3 aces in a 5-card hand from a standard deck, you would calculate: (Combinations of 4 aces taken 3 at a time × Combinations of 48 non-aces taken 2 at a time) ÷ (Combinations of 52 cards taken 5 at a time).
Are there any special properties of combinations I should know about?
Yes, combinations have several interesting properties. For instance, nCr = nC(n-r), which means the number of ways to select r items from n items is the same as the number of ways to select (n-r) items. Also, nC0 = nCn = 1, since there's only one way to select 0 items or all n items from a set of n. These properties can sometimes simplify complex combination calculations.