Permutation & Combination Calculator

About Permutation & Combination Calculator

This calculator computes permutations (nPr) and combinations (nCr) using factorial formulas. Permutations count arrangements where order matters, while combinations count selections where order doesn't matter. Both are fundamental concepts in probability theory, statistics, and combinatorics.

Formulas

Permutation (nPr):
  nPr = n! / (n-r)!

Combination (nCr):
  nCr = n! / (r! × (n-r)!)

Where:
  n! = n × (n-1) × (n-2) × ... × 2 × 1
  0! = 1 (by convention)

Example: n=5, r=3
  5P3 = 5!/(5-3)! = 120/2 = 60
  5C3 = 5!/(3!×2!) = 120/(6×2) = 10

JavaScript Implementation

// Factorial function with overflow protection
function factorial(n) {
  if (n < 0 || n > 170) return Infinity; // Beyond safe integer range
  if (n === 0 || n === 1) return 1;
  let result = 1;
  for (let i = 2; i <= n; i++) {
    result *= i;
  }
  return result;
}

// Permutation: nPr = n! / (n-r)!
function permutation(n, r) {
  if (r > n || r < 0) return 0;
  return factorial(n) / factorial(n - r);
}

// Combination: nCr = n! / (r! × (n-r)!)
function combination(n, r) {
  if (r > n || r < 0) return 0;
  return factorial(n) / (factorial(r) * factorial(n - r));
}

// Optimized combination (avoids large factorials)
function combinationOptimized(n, r) {
  if (r > n) return 0;
  if (r === 0 || r === n) return 1;
  if (r > n - r) r = n - r; // Use symmetry: nCr = nC(n-r)
  let result = 1;
  for (let i = 1; i <= r; i++) {
    result = result * (n - i + 1) / i;
  }
  return result;
}

Factorial Reference Table

Permutation vs Combination: When to Use Each

Worked Examples

Example 1: Arranging Books on Shelf
Q: How many ways to arrange 3 books from 5 different books?
A: Order matters (book positions differ), so use permutation.
   5P3 = 5!/(5-3)! = 120/2 = 60 ways

Example 2: Selecting Team Members
Q: How many ways to select 3 players from 5 players?
A: Order doesn't matter (team is same regardless of selection order).
   5C3 = 5!/(3!×2!) = 120/(6×2) = 10 ways

Example 3: Lottery Probability
Q: In a lottery picking 6 numbers from 49, what are the odds?
A: Order doesn't matter, use combination.
   49C6 = 49!/(6!×43!) = 13,983,816
   Odds of winning: 1 in 13,983,816

Example 4: Password Possibilities
Q: How many 4-digit PINs using digits 0-9 without repetition?
A: Order matters, use permutation.
   10P4 = 10!/(10-4)! = 3,628,800/720 = 5,040 PINs

Key Properties

Common Probability Applications

• Poker hands: 5-card hands from 52 cards = 52C5 = 2,598,960 possible hands
• Committee formation: Select 5 from 12 people = 12C5 = 792 ways
• Menu planning: Choose 3 appetizers from 8 = 8C3 = 56 combinations
• Route planning: Visit 4 cities in order from 10 = 10P4 = 5,040 routes
• Exam questions: Answer 5 from 8 questions = 8C5 = 56 ways to choose