Question

Find r if ¹⁴C_2r : ¹⁰C_2r–4 = 143 : 10

Answer 1

¹⁴C_2r : ¹⁰C_2r–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.