Permutations & Combinations Calculator

Permutations vs Combinations

Permutation: arrangement where ORDER matters. 'How many ways to arrange 3 books from 10?' nPr = n! / (n-r)! = 10! / 7! = 10×9×8 = 720. Combination: selection where order does NOT matter. 'How many ways to choose 3 books from 10?' nCr = n! / (r!(n-r)!) = 10! / (3!×7!) = 720/6 = 120. Memory aid: Combination lock → actually a Permutation lock (order matters for the code!). A real combination (selection) example: choosing lottery numbers. The ratio: nPr = nCr × r! (permutations = combinations × arra

Pascal's Triangle and nCr

Pascal's Triangle contains all nCr values. Row n, position r (starting from 0): nCr. Row 4: 1, 4, 6, 4, 1 = 4C0, 4C1, 4C2, 4C3, 4C4. Each entry = sum of two above. Key properties: nC0 = 1 (one way to choose nothing). nCn = 1 (one way to choose everything). nC1 = n. nCr = nC(n-r) (choosing r is same as excluding n-r). Symmetry: Pascal's Triangle is symmetric. Binomial theorem: (a+b)ⁿ = Σ nCr × aⁿ⁻ʳ × bʳ. The coefficients are the nCr values — this is why they are called binomial coefficients.

Practical Applications

Lottery (UK National Lottery): choose 6 from 59 numbers. 59C6 = 45,057,474. One in 45 million chance of winning the jackpot. Password security: 8-character password from 62 characters (a-z, A-Z, 0-9). Permutations with repetition: 62⁸ = 2.18×10¹⁴. With 10 billion password attempts per second: 6 hours to crack. 12 characters: 62¹² = 3.23×10²¹ → infeasible by brute force. Card games: 52-card deck. 5-card poker hand: 52C5 = 2,598,960. Royal flush: 4 ways (one per suit). Probability: 4/2,598,960 = 1

Factorials

n! = n × (n-1) × (n-2) × ... × 2 × 1. 0! = 1 (by definition — one way to arrange zero items). Stirling's approximation for large n: n! ≈ √(2πn) × (n/e)ⁿ. Factorials grow incredibly fast: 10! = 3,628,800. 20! = 2.43×10¹⁸. 52! = 8.07×10⁶⁷ (more than the number of atoms in Earth). This is why there are more possible orderings of a shuffled deck of cards than there are atoms on Earth — every properly shuffled deck has almost certainly never been in that exact order before.