Question

Answer the following:

Find the value of r if ⁵⁶P_r+6 : ⁵⁴P_r+3 = 30800 : 1

Answer 1

⁵⁶P_r+6 : ⁵⁴P_r+3 = 30800 : 1

∴ `(""^56"P"_("r"+6))/(""^54"P"_("r"+3)) = 30800/1`

∴ `((56!)/((56-"r"-6)!))/((54!)/((54-"r"-3)!)) = 30800`

∴ `(56!)/((56-"r"-6)!) xx ((54-"r"-3)!)/(54!) = 30800`

∴ `(56!)/((50-"r")!) xx ((51-"r")!)/(54!) = 30800`

∴ `(56 xx 55 xx 54!)/((50-"r")!) xx ((51-"r")(50-"r")!)/(54!) = 30800`

∴ 51 − r = `30800/(56 xx 55)`

∴ 51 − r = 10

∴ r = 51 − 10

∴ r = 41