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Permutations: When Order Matters

The Permutation Calculator is a tool designed to compute the number of possible ordered arrangements from a set of items. In mathematics, a permutation is a specific arrangement of objects from a set where the order in which the objects are arranged is important.

🧮 How to Use the Calculator

The calculator can handle two main types of permutation problems:

1. Permutations of all items (n!):
   • Enter the total number of items in your set into the "n" field.
   • Leave the "k" field blank.
   • Click "Calculate" to find the factorial of n (n!).
2. Permutations of a subset (nPk):
   • Enter the total number of items in the set into the "n" field.
   • Enter the number of items you are choosing to arrange into the "k" field.
   • Click "Calculate" to find the number of permutations of k items from n.

The result will be displayed along with the formula used and the steps involved in the calculation.

The Formulas Explained

1. Permutation of all 'n' items (Factorial)

When you want to find the number of ways to arrange all items in a set, you calculate the factorial of the number of items (n).

P(n, n) = n! = n × (n-1) × (n-2) × ... × 1

Example: How many ways can you arrange 3 books on a shelf? P(3,3) = 3! = 3 × 2 × 1 = 6 ways.

2. Permutation of 'k' items from 'n' (nPk)

When you are selecting and arranging a subset of 'k' items from a larger set of 'n' items, the formula is:

P(n, k) = n! / (n - k)!

Example: From a group of 5 people, how many ways can you choose a president, vice-president, and treasurer? Here, n=5 and k=3. The order matters.
P(5, 3) = 5! / (5 - 3)! = 5! / 2! = 120 / 2 = 60 ways.

💡 Frequently Asked Questions (FAQ)

What is the difference between permutations and combinations?

The key difference is order. In permutations, the order of arrangement is important (e.g., the combination lock '1-2-3' is different from '3-2-1'). In combinations, the order does not matter (e.g., a team of Ann, Bob, and Chris is the same as a team of Bob, Chris, and Ann).

What does "n!" (n factorial) mean?

n-factorial is the product of all positive integers up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, 0! = 1.