Question

Find r, if ¹²P_r–2 : ¹¹P_r–1 = 3 : 14

Answer 1

¹²P_r–2 : ¹¹P_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