Sum

Find r, if ^{12}P_{r–2} : ^{11}P_{r–1} = 3 : 14

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#### Solution

^{12}P_{r–2} : ^{11}P_{r–1} = 3 : 14

∴ `(12!)/((12 - "r" + 2)!) divide (11!)/((11 - "r" + 1)!) = 3/14`

∴ `(12!)/((14 - "r")!) xx ((12 - "r")!)/(11!) = 3/14`

∴ `(12 xx 11!)/((14 - "r")(13 - "r")(12 - "r")!) xx ((12 - "r")!)/(11!) = 3/14`

∴ `12/((14 - "r")(13 - "r")) = 3/14`

∴ (14 – r)(13 – r) = 8 × 7

Comparing on both sides, we get

∴ 14 – r = 8

∴ r = 6

Concept: Concept of Permutations - Permutations When All Objects Are Distinct

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