Question Number 80777 by zainal tanjung last updated on 14/Feb/20
$$\mathrm{If}\:\:^{{n}+\mathrm{2}} {C}_{\mathrm{8}} \::\:^{{n}−\mathrm{2}} {P}_{\mathrm{4}} =\:\mathrm{57}\::\:\mathrm{16},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{n}\:\mathrm{is}\:...... \\ $$
Answered by Joel578 last updated on 06/Feb/20
$$\frac{\left({n}+\mathrm{2}\right)!}{\mathrm{8}!\:\left({n}−\mathrm{6}\right)!}\::\:\frac{\left({n}−\mathrm{2}\right)!}{\left({n}−\mathrm{6}\right)!}\:=\:\frac{\mathrm{57}}{\mathrm{16}} \\ $$$$\Rightarrow\:\frac{\left({n}+\mathrm{2}\right)!}{\mathrm{8}!\:\left({n}−\mathrm{6}\right)!}\:×\:\frac{\left({n}−\mathrm{6}\right)!}{\left({n}−\mathrm{2}\right)!}\:=\:\frac{\mathrm{57}}{\mathrm{16}} \\ $$$$\Rightarrow\:\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right)\left({n}\right)\left({n}−\mathrm{1}\right)\:=\:\mathrm{143640} \\ $$$$\Rightarrow\:{n}\:=\:\mathrm{19} \\ $$