1. What Are Permutations?

Permutations refer to the number of possible arrangements of items where the order matters. It answers the question: "How many different ways can I arrange r items from a set of n items?"

2. How Does the Calculator Work?

The calculator uses the permutations formula:

Where:

• \( n \) — Total number of items
• \( r \) — Number of items selected
• \( ! \) — Factorial (product of all positive integers ≤ the number)

Explanation: The formula calculates all possible ordered arrangements by considering all items (n!), then dividing by the arrangements of the unselected items ((n-r)!).

3. Importance of Permutations

Details: Permutations are fundamental in probability, statistics, and combinatorics. They're used in password combinations, lottery odds, seating arrangements, and many real-world scenarios where order matters.

4. Using the Calculator

Tips: Enter positive integers where n ≥ r. For large numbers (n > 20), results will be approximate due to floating-point limitations.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between permutations and combinations?
A: Permutations consider order (ABC ≠ BAC), while combinations don't (ABC = BAC). Use combinations when order doesn't matter.

Q2: What if n = r?
A: When n = r, you're calculating all possible arrangements of all items, which is simply n! (since 0! = 1).

Q3: Can I calculate permutations for non-integer numbers?
A: No, permutations are only defined for non-negative integers. The gamma function extends factorial to real numbers, but standard permutations require whole numbers.

Q4: What's the maximum n value this calculator can handle?
A: Practically limited to n ≤ 170 due to floating-point precision. Beyond that, results become approximate.

Q5: How is this different from the multiplication principle?
A: The multiplication principle calculates arrangements with replacement, while permutations calculate arrangements without replacement.