Combination or permutation?
Ask one question: does the order matter? Picking a team of 3 from 10 students is a combination (ⁿCᵣ). Picking a captain, vice-captain and treasurer from 10 is a permutation (ⁿPᵣ), because swapping two people gives a different result.
- ⁿCᵣ = n! ÷ (r! (n − r)!). Also ⁿCᵣ = ⁿCₙ₋ᵣ, and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ.
- ⁿPᵣ = n! ÷ (n − r)! = ⁿCᵣ × r!
- With repetition: arrangements nʳ (like a 4-digit PIN, 10⁴), selections ⁿ⁺ʳ⁻¹Cᵣ.
- Circular arrangements of n different things: (n − 1)!, or (n − 1)! ÷ 2 if clockwise and anticlockwise are the same (a necklace).
- Things not all different: n! ÷ (p! q! r!…), as in the arrangements of the letters of MISSISSIPPI: 11! ÷ (4! 4! 2!) = 34,650.
JEE favourites: number of diagonals of an n-gon = ⁿC₂ − n; triangles from n points with no three in a line = ⁿC₃; ways to split 2n people into pairs = (2n)! ÷ (2ⁿ n!).
Questions students ask
What is the value of 10C3?
120. 10 × 9 × 8 ÷ (3 × 2 × 1) = 120.
What is the difference between nPr and nCr?
nPr counts arrangements where order matters; nCr counts selections where it doesn't. nPr = nCr × r!.
What is 0! equal to?
1, by definition. That's what makes nC0 = nCn = 1 work.
How many ways can 5 people sit around a round table?
(5 − 1)! = 24.