In mathematics, combinations and permutations are fundamental concepts used to count the number of ways to select items from a set. The key difference lies in whether the order of selection matters.
Permutations (Order Matters)
A permutation is an arrangement of objects in a specific order. If you have n distinct items and you want to choose and arrange r of them, the number of permutations is calculated using the formula:
P(n, r) = n! / (n - r)!
Where n! (n factorial) is the product of all positive integers up to n (e.g., 5! = 5 * 4 * 3 * 2 * 1 = 120).
Example: If you have 5 different colored balls (n=5) and you want to arrange 3 of them in a line (r=3), the order matters. The number of permutations is P(5, 3) = 5! / (5-3)! = 120 / 2 = 60.
Combinations (Order Does Not Matter)
A combination is a selection of items where the order of selection does not matter. If you have n distinct items and you want to choose r of them without regard to order, the number of combinations is calculated using the formula:
C(n, r) = n! / (r! * (n - r)!)
This is also often written as "n choose r" and denoted as (n r).
Example: If you have 5 different fruits (n=5) and you want to pick 3 to make a fruit salad (r=3), the order in which you pick them doesn't change the final salad. The number of combinations is C(5, 3) = 5! / (3! * (5-3)!) = 120 / (6 * 2) = 120 / 12 = 10.
Calculator Usage
This calculator helps you quickly compute both permutations and combinations. Simply enter the total number of items available (n) and the number of items you wish to select or arrange (r). The calculator will then provide you with the respective counts.
Total Items (n): The total number of distinct objects in your set.
Items to Select (r): The number of objects you are choosing or arranging from the set.
Both n and r must be non-negative integers, and n must be greater than or equal to r.
// Helper function to calculate factorial
function factorial(num) {
if (num < 0) {
return NaN; // Factorial is 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;
}
// Helper function to safely parse input and check for valid integers
function getValidIntegerInput(id) {
var element = document.getElementById(id);
var value = parseInt(element.value, 10);
if (isNaN(value) || value n) {
errorMessageElement.innerHTML = "Items to Select (r) cannot be greater than Total Items (n).";
return;
}
// Calculate factorials
var nFact = factorial(n);
var rFact = factorial(r);
var nMinusRFact = factorial(n – r);
if (isNaN(nFact) || isNaN(rFact) || isNaN(nMinusRFact)) {
errorMessageElement.innerHTML = "Error calculating factorials. Ensure inputs are valid.";
return;
}
// Calculate Permutations P(n, r)
var permutations = nFact / nMinusRFact;
// Calculate Combinations C(n, r)
var combinations = nFact / (rFact * nMinusRFact);
// Display results
resultElement.innerHTML = "P(n, r) = " + permutations.toLocaleString() + "" +
"C(n, r) = " + combinations.toLocaleString();
calculationDetailsElement.innerHTML =
"P(n, r) = n! / (n – r)! = " + n + "! / (" + n + " – " + r + ")! = " + nFact.toLocaleString() + " / " + nMinusRFact.toLocaleString() + " = " + permutations.toLocaleString() + "" +
"C(n, r) = n! / (r! * (n – r)!) = " + n + "! / (" + r + "! * (" + n + " – " + r + ")!) = " + nFact.toLocaleString() + " / (" + rFact.toLocaleString() + " * " + nMinusRFact.toLocaleString() + ") = " + combinations.toLocaleString();
}