Discrete Math Calculator

Discrete Math Combinations & Permutations Calculator

Use this calculator to determine the number of possible combinations or permutations for a given set of items.

function factorial(num) { if (num < 0) return NaN; // Factorial not defined for negative numbers if (num === 0 || num === 1) return 1; var result = 1; for (var i = 2; i <= num; i++) { result *= i; } return result; } function calculateCombinations() { var n = parseInt(document.getElementById("totalItemsN").value); var r = parseInt(document.getElementById("itemsToChooseR").value); var resultDiv = document.getElementById("discreteMathResult"); if (isNaN(n) || isNaN(r) || n < 0 || r n) { resultDiv.innerHTML = "'r' cannot be greater than 'n' for combinations."; return; } var nFact = factorial(n); var rFact = factorial(r); var nMinusRFact = factorial(n – r); var combinations = nFact / (rFact * nMinusRFact); resultDiv.innerHTML = "

Combinations (nCr) Result:

" + "C(" + n + ", " + r + ") = " + combinations.toLocaleString() + "" + "There are " + combinations.toLocaleString() + " ways to choose " + r + " items from a set of " + n + " items, where order does not matter."; } function calculatePermutations() { var n = parseInt(document.getElementById("totalItemsN").value); var r = parseInt(document.getElementById("itemsToChooseR").value); var resultDiv = document.getElementById("discreteMathResult"); if (isNaN(n) || isNaN(r) || n < 0 || r n) { resultDiv.innerHTML = "'r' cannot be greater than 'n' for permutations."; return; } var nFact = factorial(n); var nMinusRFact = factorial(n – r); var permutations = nFact / nMinusRFact; resultDiv.innerHTML = "

Permutations (nPr) Result:

" + "P(" + n + ", " + r + ") = " + permutations.toLocaleString() + "" + "There are " + permutations.toLocaleString() + " ways to choose and arrange " + r + " items from a set of " + n + " items, where order matters."; } .discrete-math-calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 25px; max-width: 600px; margin: 20px auto; box-shadow: 0 4px 8px rgba(0,0,0,0.05); color: #333; } .discrete-math-calculator-container h2 { color: #2c3e50; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .discrete-math-calculator-container p { line-height: 1.6; margin-bottom: 15px; } .calculator-inputs label { display: block; margin-bottom: 8px; font-weight: bold; color: #555; } .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 5px; font-size: 1em; } .calculator-buttons { text-align: center; margin-top: 20px; } .calculator-buttons button { background-color: #3498db; color: white; padding: 12px 25px; border: none; border-radius: 5px; cursor: pointer; font-size: 1.1em; margin: 0 10px; transition: background-color 0.3s ease; } .calculator-buttons button:hover { background-color: #2980b9; } .calculator-result { background-color: #eaf4f9; border: 1px solid #cce7f4; border-radius: 5px; padding: 15px; margin-top: 25px; min-height: 60px; color: #2c3e50; } .calculator-result h3 { color: #2c3e50; margin-top: 0; font-size: 1.4em; } .calculator-result p { margin-bottom: 5px; font-size: 1.1em; } .calculator-result p b { color: #e74c3c; font-size: 1.2em; }

Understanding Discrete Math: Combinations and Permutations

Discrete mathematics is a branch of mathematics dealing with discrete elements that can be counted, such as integers, graphs, and logical statements. Unlike continuous mathematics (which deals with real numbers and continuous functions), discrete math focuses on structures that are fundamentally separate or distinct.

What are Combinations?

In discrete mathematics, a combination is a selection of items from a larger set where the order of selection does not matter. It answers the question: "How many ways can you choose 'r' items from a set of 'n' items, without regard to the order of selection?"

The formula for combinations is:

C(n, r) = n! / (r! * (n-r)!)

Where:

• n is the total number of items available.
• r is the number of items to choose.
• ! denotes the factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1).

Example: If you have 10 different books and you want to choose 3 to read, how many different sets of 3 books can you choose? Here, n=10, r=3. The order in which you pick the books doesn't change the set of books you end up with. Using the calculator with n=10 and r=3 for combinations would yield 120.

What are Permutations?

A permutation is an arrangement of items from a larger set where the order of selection (or arrangement) does matter. It answers the question: "How many ways can you choose and arrange 'r' items from a set of 'n' items?"

The formula for permutations is:

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

Where:

• n is the total number of items available.
• r is the number of items to choose and arrange.
• ! denotes the factorial.

Example: Imagine a race with 10 runners. How many different ways can the gold, silver, and bronze medals be awarded? Here, n=10, r=3. The order matters because coming in first is different from coming in second. Using the calculator with n=10 and r=3 for permutations would yield 720.

How to Use This Calculator:

1. Total Number of Items (n): Enter the total count of distinct items you have. This must be a non-negative integer.
2. Number of Items to Choose (r): Enter the count of items you wish to select or arrange from the total set. This must be a non-negative integer and cannot be greater than 'n'.
3. Calculate Combinations (nCr): Click this button if the order of selection does NOT matter.
4. Calculate Permutations (nPr): Click this button if the order of selection DOES matter.
5. The result will be displayed below the buttons, showing the calculated number of combinations or permutations along with a brief explanation.