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Permutations & Combinations
· Formula🧮 Formula Reference Sheet
nPr = n! ÷ (n−r)! [ordered arrangements] nCr = n! ÷ (r! × (n−r)!) [unordered selections] nCr = nC(n−r) [symmetry property] nPn = n! [all items arranged] Circular: (n−1)! [around a table] nC0 = nCn = 1 [trivial cases]
⚡ Example per formula
nPr = n! ÷ (n−r)! [ordered arrangements]
↳5P3 = 5! / 2! = 60 (e.g. arrange 3 of 5 books)
nCr = n! ÷ (r! × (n−r)!) [unordered selections]
↳6C3 = 6! / (3!×3!) = 20 (committee choices)
nCr = nC(n−r) [symmetry property]
↳6C3 = 6! / (3!×3!) = 20 (committee choices)
nPn = n! [all items arranged]
↳4 people in 4 chairs → 4! = 24
Circular: (n−1)! [around a table]
↳8 around table → (8−1)! = 7! = 5040
nC0 = nCn = 1 [trivial cases]
↳7C0 = 7C7 = 1
✏️ Worked Example
In how many ways can the letters of the word "MATH" be arranged?
👁 Show step-by-step solution
4 distinct letters, all arranged: 4! = 4 × 3 × 2 × 1 = 24 ✅ Answer: 24
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Practice Exercises →
5 graded problems · AI checks each step