Question Number 180773 by Acem last updated on 17/Nov/22
$${How}\:{many}\:{triangles}\:{can}\:{be}\:{formed}\:{from} \\ $$$$\:{non}−{adjacent}\:{vertices}\:{of}\:{a}\:{regular}\:{polygon} \\ $$$$\:{that}\:{it}\:{angle}\:{is}\:\mathrm{177}^{\:°} \:? \\ $$
Answered by mr W last updated on 17/Nov/22
$${n}=\frac{\mathrm{360}}{\mathrm{180}−\mathrm{177}}=\mathrm{120}\:{vertices} \\ $$$${we}\:{can}\:{form} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}×{n}×\left({C}_{\mathrm{2}} ^{{n}−\mathrm{3}} −\left({n}−\mathrm{4}\right)\right) \\ $$$$=\frac{{n}\left({n}−\mathrm{4}\right)\left({n}−\mathrm{5}\right)}{\mathrm{6}} \\ $$$$=\mathrm{266}\:\mathrm{800}\:{triangles} \\ $$
Commented by Acem last updated on 17/Nov/22
$${Morning}\:{Sir}!,\:{the}\:{number}\:{is}\:{rather}\:{close}\:{to} \\ $$$$\:{the}\:{correct}\:{answer},\:{if}\:{we}\:{apply}\:{the}\:{formula}\:{above} \\ $$$$\:{to}\:{a}\:{hexagon}\:{or}\:{otherwise}\:{we}\:{wouldn}'{t}\:{get}\:{the} \\ $$$$\:{real}\:{answer}. \\ $$
Commented by mr W last updated on 17/Nov/22
$${i}\:{see}\:{my}\:{mistake}\:{now}.\:{i}'{ll}\:{fix}\:{it}. \\ $$$${the}\:{vertrices}\:{of}\:{triangle}\:{should}\:{not} \\ $$$${be}\:{adjacent}\:{vertrices}\:{of}\:{the}\:{polygon}. \\ $$$${i}\:{didn}'{t}\:{follow}\:{this}\:{correctly}. \\ $$
Commented by Acem last updated on 17/Nov/22
$${Yes}\:{Sir},\:{you}\:{will}\:{do}\:{it}!\:{thank}\:{you} \\ $$$$ \\ $$$$\left.\:{Note}:\:{The}\:{number}\:\in\:\right]\mathrm{265}\:\mathrm{349},\:\mathrm{266}\:\mathrm{801}\left[\right. \\ $$
Commented by mr W last updated on 17/Nov/22
$${i}\:{got}\:\mathrm{266800}.\:{please}\:{check}\:{my}\:{fixed}\: \\ $$$${solution}. \\ $$
Commented by Acem last updated on 17/Nov/22
$${Shakehands}! \\ $$$${Whatever}\:{Regular}\:{Polygon\begin{cases}{{C}_{\mathrm{3}} ^{\:{n}} \:−\:{n}\left({n}−\mathrm{3}\right)}\\{\:\frac{{n}\left({n}−\mathrm{4}\right)\left({n}−\mathrm{5}\right)}{\mathrm{6}}}\end{cases}} \\ $$$$ \\ $$$$\ast\:{If}\:{we}\:{want}\:{to}\:{count}\:{only}\:{the}\:{triangles}\:{that}\:{are} \\ $$$$\:{formed}\:{from}\:{adjacent}\:{vertices},\:{apply}\:\boldsymbol{{n}}\left(\boldsymbol{{n}}−\mathrm{3}\right) \\ $$$$ \\ $$