Question

Find r if `""^14"C"_(2"r"): ""^10"C"_(2"r" - 4)` = 143:10

Answer 1

`""^14"C"_(2"r"): ""^10"C"_(2"r" - 4)` = 143:10

`∴(14!)/(2"r"!(14-2"r")!)÷(10!)/((2"r" - 4)!(14-2"r")!) = 143/10`

∴ `(14!)/(2"r"!(14 - 2"r")!) xx ((2"r" - 4)!(14-2"r")!)/(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)!(14-2"r")!)xx((2"r"-4)!(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`

∴ 2r(2r – 1) . (2r – 2) (2r – 3) = 14 × 12 × 10

∴ 2r(2r – 1) . (2r – 2) (2r – 3) = 8 × 7 × 6 × 5

Comparing on both sides, we get
∴ r = 4