Permutation Calculator

About Permutation Calculator

Permutations count the number of ways to arrange r items from n items where order matters.

How to use this calculator

1. Enter the total number of available items, \(n\).
2. Enter how many items you want to arrange, \(r\).
3. Make sure \(r \le n\), because you cannot arrange more items than you have.
4. Read the value of \(P(n,r)\), which gives the number of ordered arrangements.

The formula explained

The permutation formula \(P(n,r)=\frac{n!}{(n-r)!}\) computes the number of ways to choose and arrange \(r\) items from \(n\) distinct items. It counts ordered selections, so changing the order changes the result.

• n = the total number of distinct items available
• r = the number of items selected and arranged
• ! = factorial, the product of all positive integers from that number down to 1

Step by step method

1. Start with the permutation formula, \(P(n,r)=\frac{n!}{(n-r)!}\).
2. Expand the factorials or cancel matching factors to simplify the expression.
3. Compute the remaining product to get the number of ordered arrangements.

Worked example

Problem. How many 3-letter arrangements can be made from the letters A, B, C, D, and E if no letter is repeated?

1. Here, \(n=5\) and \(r=3\), so use \(P(5,3)=\frac{5!}{(5-3)!}\).
2. Simplify the denominator: \(P(5,3)=\frac{5!}{2!}=5\times4\times3=60\).
3. There are 60 ordered arrangements.

Tips and common mistakes

• Order matters in permutations, so ABC and ACB are different arrangements.
• Check that \(r\) is not greater than \(n\), and remember that repeating items changes the problem type.

Frequently asked questions