Permutation and Combination
Calculate permutations P(n,r) and combinations C(n,r) easily online. Enter n total items and r items to choose for instant P(n,r) and C(n,r) results with a chart.
About This Calculator
The Permutation and Combination Calculator computes two fundamental values in combinatorics: the number of permutations P(n,r) and the number of combinations C(n,r) for a given set of n distinct items from which you select r items. Whether you are a student learning probability and counting principles, a teacher preparing combinatorics lessons, or a data scientist calculating binomial coefficients, this tool provides instant, accurate results with an interactive comparison chart.
Permutations count the number of ways to arrange r items selected from n distinct items when order matters. The formula is P(n,r) = n! / (n-r)!. For example, selecting a president, vice-president, and secretary from a group of 10 people gives P(10,3) = 720 possible leadership arrangements. Combinations count selections where order does not matter, using the formula C(n,r) = n! / (r! x (n-r)!). Choosing a 3-person committee from the same 10 people gives C(10,3) = 120 possible committees -- six times fewer because each committee can be arranged in 3! = 6 ways.
The calculator uses efficient iterative algorithms for both nPr and nCr to avoid computing large intermediate factorials. The combination function uses the symmetry property C(n,r) = C(n,n-r) to minimize iterations, selecting the smaller of r and n-r for faster computation. Results are formatted with locale-aware thousand separators for readability, and a bar and line chart visualize how permutation and combination values change across different r choices from 1 to n.
Regional Notes
India (IN): Permutations and combinations are core topics in CBSE and state board mathematics curricula from Class 11 onward, appearing in probability, binomial theorem, and JEE Main/Advanced exam problems. The calculator uses standard nPr and nCr notation matching NCERT textbooks.
United States (US): Combinatorics is taught in high school Algebra 2, Precalculus, and AP Statistics. The calculator supports the standard notation P(n,r) and C(n,r) (or nCr) used in Common Core math and SAT/ACT probability sections.
United Kingdom (UK): Permutations and combinations are covered in A-Level Mathematics (Pure and Statistics) under probability and counting principles. The binomial coefficient notation C(n,r) matches the Edexcel, AQA, and OCR specifications.
Frequently Asked Questions
What is the difference between a permutation and a combination?
A permutation considers the order of selection as important (arranging items in sequence), while a combination treats order as irrelevant (selecting a group). For example, picking 3 winners for 1st, 2nd, 3rd place is a permutation; picking a 3-person committee is a combination. Permutations always yield a larger number than combinations for the same n and r values.
What formula does the permutation calculator use?
The permutation formula used is P(n,r) = n! / (n-r)!, where n is the total number of items and r is the number of items to arrange. This counts all possible ordered arrangements of r items selected from n distinct items.
What formula does the combination calculator use?
The combination formula used is C(n,r) = n! / (r! x (n-r)!), also known as the binomial coefficient or nCr. This counts all possible unordered selections of r items from n distinct items. The result represents the number of distinct subsets of size r.
What happens when r is greater than n?
When r exceeds n, the calculation is invalid because you cannot choose more items than are available. The calculator displays an error message: Items to choose (r) cannot exceed total items (n). Adjust your inputs so that r is less than or equal to n.
Can I use this calculator for probability problems?
Yes. Permutations and combinations are fundamental to probability. Use the combination result as the denominator for classical probability calculations (favorable outcomes divided by total outcomes). For example, the probability of winning the lottery uses combinations to count all possible ticket combinations.
What is the maximum n value supported by this calculator?
The calculator uses double-precision floating-point arithmetic, supporting n values up to approximately 170 before factorial overflow occurs. For most practical problems involving combinatorics, n values under 100 produce accurate integer results displayed with comma separators.
Is this permutation combination calculator free to use?
Yes, this calculator is completely free to use with no registration required. You can also share your calculation results by copying the URL, which preserves your n and r values for easy reference.