Combination and Permutation Calculator

What is Combination and Permutation?

Combinations and permutations are fundamental concepts in combinatorics. A permutation refers to the arrangement of objects in a specific order. For example, the sequence of characters in a password is a permutation where order matters. A combination, however, focuses on selecting objects without considering order, such as choosing members for a committee.

How to Calculate Combination

Combinations are calculated using the formula: \(\displaystyle C(n, k) = \frac{n!}{k!(n-k)!}\) Here, \(n\) is the total number of items, \(k\) is the number of items to choose, and "!" denotes factorial. For example, choosing 3 students from a group of 10 is: \(\displaystyle C(10, 3) = \frac{10!}{3!7!} = 120\) ways.

How to Calculate Permutation

Permutations use the formula: \(\displaystyle P(n, k) = \frac{n!}{(n-k)!}\) Order matters here. For instance, arranging 4 books out of 7 on a shelf is: \(\displaystyle P(7, 4) = \frac{7!}{3!} = 840\) ways.

Applications of Combination and Permutation in Real Life

• Permutations: Password generation, race rankings, and scheduling events.
• Combinations: Lottery number selection, forming project teams, and statistical sampling.

In cryptography, permutations ensure secure encryption. In biology, combinations help analyze gene pairs.

Our Combination & Permutation Calculator

While understanding the math is key, our tool simplifies calculations. Enter \(n\) and \(k\), and it handles factorials instantly. For example, confirm \(C(10, 3)\) or \(P(7, 4)\) in seconds. Use it to check your work or explore larger values!