Combination Permutation Calculator

Combination & Permutation Calculator

function factorial(num) { if (num < 0) return NaN; if (num === 0 || num === 1) return 1; var result = 1; for (var i = 2; i <= num; i++) { result *= i; } return result; } function calculateCombinationsPermutations() { var totalItems = parseInt(document.getElementById('totalItems').value); var itemsToChoose = parseInt(document.getElementById('itemsToChoose').value); var permutationsResultElement = document.getElementById('permutationsResult'); var combinationsResultElement = document.getElementById('combinationsResult'); if (isNaN(totalItems) || isNaN(itemsToChoose) || totalItems < 0 || itemsToChoose totalItems) { permutationsResultElement.textContent = "0 (k cannot be > n)"; combinationsResultElement.textContent = "0 (k cannot be > n)"; return; } var nFactorial = factorial(totalItems); var kFactorial = factorial(itemsToChoose); var nMinusKFactorial = factorial(totalItems – itemsToChoose); var permutations = nFactorial / nMinusKFactorial; var combinations = nFactorial / (kFactorial * nMinusKFactorial); permutationsResultElement.textContent = permutations.toLocaleString(); combinationsResultElement.textContent = combinations.toLocaleString(); } // Initial calculation on page load window.onload = calculateCombinationsPermutations;

Understanding Combinations and Permutations

Combinations and permutations are fundamental concepts in probability and statistics, used to determine the number of ways to select or arrange items from a larger set. While often confused, the key difference lies in whether the order of selection matters.

What is a Permutation?

A permutation is an arrangement of items where the order of selection is important. If you're arranging objects, the sequence in which they appear creates a distinct permutation. For example, if you have three letters A, B, C, the permutations of choosing two letters are AB, BA, AC, CA, BC, CB. Notice that AB is different from BA because the order has changed.

The formula for calculating permutations of choosing 'k' items from a set of 'n' items is:

P(n, k) = n! / (n – k)!

Where 'n!' (n factorial) is the product of all positive integers up to n (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120). By definition, 0! = 1.

Permutation Example:

Imagine a race with 10 runners. How many different ways can the gold, silver, and bronze medals be awarded? Here, the order matters (gold is different from silver). So, n = 10 (total runners) and k = 3 (medal positions).

P(10, 3) = 10! / (10 – 3)! = 10! / 7! = 10 × 9 × 8 = 720 ways.

What is a Combination?

A combination is a selection of items where the order of selection does not matter. If you're simply choosing a group of objects, the arrangement within that group doesn't create a new combination. Using the letters A, B, C again, the combinations of choosing two letters are AB, AC, BC. Here, AB is considered the same as BA because it's the same group of letters.

The formula for calculating combinations of choosing 'k' items from a set of 'n' items is:

C(n, k) = n! / (k! * (n – k)!)

Combination Example:

Suppose you have 10 friends, and you want to invite 3 of them to a party. How many different groups of 3 friends can you invite? The order in which you invite them doesn't matter; a group of John, Mary, and Sue is the same as Sue, John, and Mary. So, n = 10 (total friends) and k = 3 (friends to invite).

C(10, 3) = 10! / (3! * (10 – 3)!) = 10! / (3! * 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 groups.

How to Use the Calculator

Our Combination & Permutation Calculator simplifies these calculations for you:

1. Total Number of Items (n): Enter the total number of distinct items available in your set.
2. Number of Items to Choose (k): Enter the number of items you want to select or arrange from the total set.
3. Click the "Calculate" button.

The calculator will instantly display both the number of possible permutations and combinations based on your inputs, helping you quickly solve problems in probability, statistics, and various real-world scenarios.