Permutations & Combinations Calculator
Calculate the number of ways to arrange or select items from a set. Permutations care about order; combinations do not. Enter n and r to get both values.
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What's the difference between Permutations and 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! shows that permutations equal combinations multiplied by the number of ways to arrange each selected group, which is exactly why permutations always count at least as many outcomes as combinations for the same n and r.
What should I know about 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.
What should I know about 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 = roughly 1 in 650,000, showing just how rare a royal flush genuinely is even across a large number of hands dealt.
What do I need to know about 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.