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Calculate permutations (nPr) and combinations (nCr) with step-by-step factorial breakdowns. Includes a quick reference table for common values.
Permutations (nPr) count the number of ordered arrangements of r items from n, while combinations (nCr) count the number of unordered selections. Both are fundamental concepts in combinatorics and probability.
Permutations (nPr) — order matters
720
10P3 = 10! / 7!
Combinations (nCr) — order doesn't matter
120
10C3 = 10! / (3! × 7!)
Step-by-Step Breakdown
10! = 3,628,800
3! = 6
(10 − 3)! = 7! = 5,040
nPr = 3,628,800 / 5,040 = 720
nCr = 3,628,800 / (6 × 5,040) = 120
| n, r | nPr | nCr |
|---|---|---|
| 5, 2 | 20 | 10 |
| 5, 3 | 60 | 10 |
| 10, 3 | 720 | 120 |
| 10, 5 | 30,240 | 252 |
| 52, 5 | 311,875,200 | 2,598,960 |
| 26, 2 | 650 | 325 |
Formula
nPr = n! / (n−r)! | nCr = n! / (r! × (n−r)!)n = Total number of items in the set
r = Number of items being chosen or arranged
n! = n factorial — the product of all positive integers up to n
Worked Example
How many 5-card poker hands from a 52-card deck?
Did you know? The number of ways to arrange a standard 52-card deck is 52! ≈ 8.07 × 10⁶⁷ — more than the estimated number of atoms in the observable universe (~10⁸⁰), meaning no two properly shuffled decks have likely ever been in the same order (source: American Mathematical Society).
Sources
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