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Question
Find r if 14C2r : 10C2r–4 = 143 : 10
Solution
14C2r : 10C2r–4 = 143 : 10
∴ `(14!)/(2"r"!(14 - 2"r")!) xx ((2"r" - 4)!(10 - 2"r" + 4)!)/(10!) = 143/10`
∴ `(14 xx 13 xx 12 xx 11 xx 10!)/(2"r"(2"r" - 1)(2"r" - 2)(2"r" - 3)(2"r" - 4)! xx (14 - 2"r")!) xx ((2"r" - 4)! xx (14 - 2"r")!)/(10!) = 143/10`
∴ `(14 xx 13 xx 12 xx 11)/(2"r"(2"r" - 1)(2"r" - 2)(2"r" - 3)) = 143/10`
∴ `(14 xx 12 xx 143)/(2"r"(2"r" - 1)(2"r" - 2)(2"r" - 3)) = 143/10`
∴ 14 × 12 × 10 = 2r(2r – 1)(2r – 2)(2r – 3)
∴ 2r(2r – 1)(2r – 2)(2r – 3) = 1680
∴ 2r(2r – 1)(2r – 2)(2r – 3) = 8 × 7 × 6 × 5
∴ 2r(2r – 1)(2r – 2)(2r – 3) = 8(8 – 1)(8 – 2)(8 – 3)
∴ 2r = 8 (by observation)
∴ r = 4.
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