Combination Calculator
Calculate combinations (nCr) or permutations (nPr) for given n and r. Useful for probability and statistics.
Result
Please check your inputs.
How to Use This Tool
Enter the total number of items (n) in the first input field. This is the size of your set.
Enter the number of items to choose (r) in the second field.
Choose whether you want to calculate combinations (nCr, order does not matter) or permutations (nPr, order matters) using the toggle or dropdown.
Click the 'Calculate' button.
View the result displayed instantly along with a step-by-step breakdown of the formula used.
What Is Combination Calculator?
The Combination Calculator is a simple yet powerful educational tool designed to quickly compute combinations (nCr) and permutations (nPr) for any given n and r values. In probability and statistics, these calculations are fundamental for determining the number of ways to choose or arrange items from a larger set. Understanding combinations and permutations helps in fields like data science, game theory, genetics, and everyday decision-making — from lottery odds to team selection. This tool eliminates manual calculation errors and provides clear, instant results, making complex math accessible to students, educators, and professionals alike.
Formula
For combinations (nCr): C(n, r) = n! / [r! * (n - r)!] — where n! means n factorial (multiply all numbers from n down to 1). For permutations (nPr): P(n, r) = n! / (n - r)! — the same factorial but without dividing by r!. In plain English: combinations count selections where order doesn't matter (e.g., picking 3 fruits from a basket), while permutations count arrangements where order does matter (e.g., ranking the first 3 finishers in a race).
Tips for Best Results
🔢 Double-check that r is not larger than n — combinations and permutations are only defined when r ≤ n.
📝 Use the nCr option when dealing with groups or subsets (e.g., hand of cards), and nPr for rankings or sequences (e.g., passwords).
📱 For large n (like n > 100), the tool uses efficient integer arithmetic to avoid overflow — no more scientific notation surprises.
🎯 Save time by using this calculator to verify your homework or real-world probability problems instead of manual factorial crunching.
Frequently Asked Questions
What is the difference between nCr and nPr?
nCr (combinations) counts selections where order does not matter — for example, choosing 3 team members from a group of 10. nPr (permutations) counts arrangements where order is important — like assigning different roles to 3 people from the same group. The formula for nPr is larger because it treats each ordering as a unique outcome.
Can I use this calculator for very large numbers like n=500?
Yes, the Combination Calculator is built to handle large values of n and r using precise integer arithmetic. However, extremely large results (e.g., n=500, r=250) may be displayed in scientific notation or as a special large-number string. For most practical purposes up to n=1000, the tool works reliably.
Why does the calculator show a zero for some inputs?
If you see a result of zero, it usually means r is greater than n — which is mathematically invalid for both combinations and permutations (you cannot choose more items than you have). Alternatively, if both n and r are zero, the combination is defined as 1 (exactly one way to choose nothing). Always ensure 0 ≤ r ≤ n.