Permutations and Combinations

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Factorial notation n! (read as 'n factorial') is the product of first n natural numbers. n! = 1 × 2 × 3 × ... × (n-1) × n Special cases: 0! = 1, 1! = 1 Example: 5! = 1 × 2 × 3 × 4 × 5 = 120 Recursive

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If event E can occur in m ways and event F can occur in n ways (and both events cannot occur simultaneously), then either event E or F can occur in (m + n) ways. Example: If there are 3 physics books

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If an event can occur in m different ways, and following this, another event can occur in n different ways, then the total number of ways both events can occur in succession is m × n. Example: If ther

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A permutation is an arrangement of objects in a definite order, taking some or all objects at a time. Order matters in permutations. Example: The letters A, B, C can be arranged as ABC, ACB, BAC, BCA,

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ⁿPᵣ = n!/(n-r)! where n ≥ r ≥ 0 Alternatively: ⁿPᵣ = n(n-1)(n-2)...(n-r+1) Example: ⁵P₃ = 5!/(5-3)! = 5!/2! = 5×4×3 = 60 When r = n: ⁿPₙ = n!/0! = n!

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When repetition is allowed, the number of permutations of n distinct objects taken r at a time is: nʳ Example: How many 3-digit numbers can be formed using digits 1,2,3 with repetition? Answer: 3³ =

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HONEST has 6 distinct letters. We need to arrange 4 letters from these 6. Using permutation formula: ⁶P₄ = 6!/(6-4)! = 6!/2! = 6×5×4×3 = 360 Therefore, 360 four-letter words can be formed.

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When n objects include p₁ objects of one kind, p₂ objects of second kind, ..., pₖ objects of kth kind: Formula: n!/(p₁! × p₂! × ... × pₖ!) Example: Letters of LETTER L-1, E-2, T-2, R-1 (total 6 lett