Permutations, Combination and Binomial Expansion

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If there are m distinct things in Group 1 and n distinct things in Group 2, then selection of one thing from the combined groups can be done in m + n ways. The keyword 'OR' indicates addition. Example

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If the first operation can be done in m ways and the second operation can be done in n ways, then both operations together can be done in m × n ways. The keyword 'AND' indicates multiplication. Exampl

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n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1 It is the product of all natural numbers from 1 to n. Examples: 5! = 5 × 4 × 3 × 2 × 1 = 120 3! = 3 × 2 × 1 = 6 1! = 1 0! = 1 (by definition)

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Permutation is the arrangement of r distinct things out of n distinct things where ORDER MATTERS. Formula: ⁿPᵣ = n!/(n-r)! where n ≥ r ≥ 0 Example: Arranging 3 students out of 5 students in 3 chairs =

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ⁿP₀ = 1 (no arrangements) ⁿPₙ = n! (all things arranged) ⁿP₁ = n (one thing from n) ⁿPₙ₋₁ = n! (leave one thing out) Examples: ⁵P₀ = 1, ⁵P₅ = 5! = 120, ⁵P₁ = 5, ⁵P₄ = 5! = 120

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Combination is the selection of r things out of n distinct things where ORDER DOES NOT MATTER. Formula: ⁿCᵣ = n!/[r!(n-r)!] where n ≥ r ≥ 0 Example: Selecting 3 friends out of 5 friends = ⁵C₃ = 5!/(3!

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ⁿPᵣ = ⁿCᵣ × r! OR ⁿCᵣ = ⁿPᵣ/r! This means: For each combination of r things, there are r! different arrangements (permutations). Example: ⁵C₃ = 10 combinations, but ⁵P₃ = 10 × 3! = 60 permutations

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ⁿC₀ = 1 (select nothing) ⁿCₙ = 1 (select everything) ⁿC₁ = n (select one thing) ⁿCₙ₋₁ = n (leave one thing) ⁿCᵣ = ⁿCₙ₋ᵣ (symmetry property) Example: ⁵C₂ = ⁵C₃ = 10