Understanding Counting Choices
Combinations and permutations solve different counting questions. A permutation counts ordered arrangements. A combination counts selections where order does not matter. This calculator keeps both methods visible, so you can compare answers before using them in probability, planning, coding, finance, gaming, or classroom work.
Why Order Matters
Order changes the result when positions, ranks, passwords, seating plans, or prize places are involved. The same three people can create many lineups when first, second, and third positions matter. Order does not change a result when you only pick a team, choose toppings, or select committee members. The tool lets you switch methods without rebuilding your inputs.
Advanced Options Included
Use the main fields for n and r. Here, n is the total number of available items. The value r is the number selected or arranged. You can also test repetition. Arrangements with repetition are useful for PIN codes, product codes, license patterns, and quiz attempts. Combinations with repetition are useful when items can be chosen again, such as scoops, bundles, and inventory groups.
Repeated Item Support
Some arrangements contain identical items. Words like LEVEL, ARRAY, or SUCCESS have repeated letters. A normal factorial would overcount those cases. Enter repeated group sizes, separated by commas, to reduce the result correctly. For SUCCESS, the repeated counts include 3 for S and 2 for C.
Using Results Safely
Large counting problems grow very fast. Even small inputs can create results with many digits. Use exact values for reporting. Use the comparison rows for checking logic. Review the formula shown beside each answer. Export the table when you need records for lessons, audits, reports, or later review.
Practical Uses
This calculator is useful for lottery analysis, schedule design, game builds, tournament brackets, password spaces, sample selection, and quality testing. It can also support statistics homework. The example table gives starting cases. Change one input at a time. Then watch how each method responds. That habit makes the difference between guessing and understanding. When order, repetition, or identical items change, the final answer can shift dramatically.
For best accuracy, define the counting question first. Ask whether sequence matters. Ask whether items return to the pool. Ask whether repeated items should merge together clearly.