What is a different combinations calculator?
A different combinations calculator helps you quickly count how many unique ways items can be selected or arranged. It is useful when you need exact counts for decision-making, probability, planning, scheduling, coding, statistics, and game design. Instead of manually expanding formulas, you enter values and get immediate results.
This page supports four common counting modes: standard combinations, permutations, combinations with repetition, and total subsets. Together, these cover most practical counting scenarios.
Core counting modes explained
1) Combinations (nCr)
Choose r items from n items when order does not matter. Example: choosing 3 students for a committee from 10 students.
nCr = n! / (r! (n-r)!)
2) Permutations (nPr)
Choose and arrange r items from n items when order does matter. Example: awarding gold, silver, and bronze from 10 finalists.
nPr = n! / (n-r)!
3) Combinations with repetition
Choose r items from n types when repeating a type is allowed. Example: selecting 5 donuts from 3 flavors where you can pick the same flavor multiple times.
(n + r - 1)Cr
4) All possible subsets
Count every possible subset of a set of size n, including the empty set and the full set. This is common in feature selection and logic exploration.
2^n
Quick examples
- nCr: 10 choose 3 = 120
- nPr: 10 permute 3 = 720
- Repetition: n=3, r=5 gives 21 combinations
- Subsets: n=8 gives 256 total subsets
When to use each mode
- Use combinations for committees, bundles, or groups.
- Use permutations for rankings, seat orders, or codes where position matters.
- Use repetition mode for “pick with replacement” style selections.
- Use subsets for yes/no inclusion decisions across many items.
Common mistakes to avoid
- Mixing up combinations and permutations.
- Using r > n in modes where that is invalid (nCr and nPr).
- Entering decimals instead of whole numbers.
- Forgetting whether repeated picks are allowed.
Practical applications
Probability and statistics
Hypergeometric probability, sampling, and experimental design frequently require nCr terms.
Business and operations
Scheduling teams, packaging product options, and building candidate shortlists all rely on combination counts.
Software engineering and data science
Feature subset exploration, A/B variant planning, and test-case coverage often depend on subset and permutation counts.
Final note
This calculator returns exact integer results using big-number arithmetic, so it can handle values larger than standard floating-point calculators. For very large inputs, results can have many digits, so choose realistic ranges for easier interpretation.