Permutation & Combination Calculator – User Guide

Introduction

The Permutation & Combination Calculator is a tool designed to help you compute permutations and combinations for given values of n (total items) and r (items to choose). It supports calculations with and without repetition.

Understanding Permutations and Combinations

Permutation

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. The number of ways to arrange r items from a set of n items is calculated differently based on whether repetition is allowed.

Permutation Without Repetition

The formula for permutations without repetition is:

Where:

• n is the total number of items.
• r is the number of items being chosen.
• n! is the factorial of n.

Permutation With Repetition

The formula for permutations with repetition is:

Combination

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

Combination Without Repetition

The formula for combinations without repetition is:

Combination With Repetition

The formula for combinations with repetition is:

How to Use the Calculator

1. Open the calculator in your web browser.
2. Enter the value of n (total items) in the provided input field. It must be a non-negative integer less than or equal to 170.
3. Enter the value of r (items to choose) in the provided input field. It must be a non-negative integer less than or equal to 170.
4. If you wish to allow repetition of items in permutations and combinations, check the Allow Repetition checkbox.
5. Click the Calculate button to compute the results.
6. The calculator will display:
   • The calculated permutation and combination values.
   • The formulas used for the calculations based on your input.

Important Notes

• Non-negative Integers: Both n and r must be non-negative integers.
• n ≥ r: When repetition is not allowed, n must be greater than or equal to r.
• Input Limits: Due to computational limitations, n and r should be less than or equal to 170 to prevent overflow errors.
• Factorial (!): The factorial of a number is the product of all positive integers less than or equal to that number. By definition, 0! = 1.

Examples

Example 1: Permutation Without Repetition

Calculate the number of ways to arrange 3 items out of 5 without repetition.

Input: n = 5, r = 3, Repetition = No

Calculation:

Result: 60 permutations.

Example 2: Combination With Repetition

Calculate the number of ways to choose 4 items out of 6 with repetition.

Input: n = 6, r = 4, Repetition = Yes

Calculation:

Result: 126 combinations.

Understanding Repetition

Repetition allows the same item to be chosen more than once. Here’s how it affects calculations:

• Permutations with Repetition: Each of the r positions can be filled by any of the n items. Therefore, the total number of permutations is n^r.
• Combinations with Repetition: The formula adjusts for the repetition by considering the combinations of n + r – 1 items taken r at a time.

Troubleshooting

If you encounter issues or error messages:

• Ensure that n and r are non-negative integers.
• Check that n is greater than or equal to r when repetition is not allowed.
• Avoid using values greater than 170 for n or r to prevent overflow errors.
• Make sure your browser supports JavaScript and that it is enabled.

Contact and Support

If you have questions, feedback, or need assistance with the calculator, please contact the support team at support@example.com.

Disclaimer

This calculator is intended for educational purposes. While we strive for accuracy, we cannot guarantee the results for all possible inputs. Please verify the results independently if using them for critical calculations.