Question

Find n and r if `""^"n""C"_("r" - 1): ""^"n""C"_"r": ""^"n""C"_("r" + 1)` = 20:35:42

Answer 1

`""^"n""C"_("r" - 1): ""^"n""C"_"r": ""^"n""C"_("r" + 1)` = 20:35:42

∴ `""^"n""C"_("r" - 1):""^"n""C"_"r" = 20/35`

∴ `("n"!)/(("r" - 1)!["n" - ("r" - 1)]!)÷("n"!)/("r"!("n" - "r")!) = 4/7`

∴ `("n"!)/(("r" - 1)!("n"+1-"r")!) xx ("r"!("n"-"r")!)/("n"!) = 4/7`

∴ `("n"!)/(("r" - 1)!("n"+1-"r")!) xx ("r"("r" - 1)!("n"-"r")!)/("n"!) = 4/7`

∴ `("r"("n"-"r")!)/(("n" + 1 - "r" )("n"-"r")!) = 4/7`

∴ 7r = 4 (n + 1 − r)
∴ 7r = 4n + 4 − 4r
∴ 11r = 4n + 4  ...(I)

Also, `""^"n""C"_"r": ""^"n""C"_("r" + 1)` = 35:42

∴ `(""^"n""C"_"r")/(""^"n""C"_("r" + 1)) = 35/42`

∴ `("n"!)/("r"!("n" - "r")!)÷("n"!)/(("r" + 1)!("n" - "r" - 1)!) = 5/6`

∴ `("n"!)/("r"!("n" - "r")!) xx (("r" + 1)!("n" - "r" - 1)!)/("n"!) = 5/6`

∴ `(("r" + 1)!("n" - "r" - 1)!)/("r"!("n" - "r")("n" - "r" - 1)!) = 5/6`

∴ `("r" + 1)/("n" - "r")=5/6`

∴ 6r + 6 = 5n − 5r

11r = 5n − 6

∴ 4n + 4 = 5n − 6 .....[From (I)]
∴ n = 10
∴ 11r = 4(10) + 4 .......[From (I)]
= 44
∴ r = 4