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Question Number 21683 by Tinkutara last updated on 30/Sep/17
$$\mathrm{Suppose}\:{A}_{\mathrm{1}} {A}_{\mathrm{2}} ...{A}_{\mathrm{20}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{20}-\mathrm{sided}\:\mathrm{regular} \\ $$$$\mathrm{polygon}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{non}-\mathrm{isosceles} \\ $$$$\left(\mathrm{scalene}\right)\:\mathrm{triangles}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{whose} \\ $$$$\mathrm{vertices}\:\mathrm{are}\:\mathrm{among}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{polygon}\:\mathrm{but}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}? \\ $$
Commented by alex041103 last updated on 30/Sep/17
$${Isn}'{t}\:{it}\:\mathrm{780}? \\ $$
Commented by Tinkutara last updated on 03/Oct/17
$$\mathrm{No}.\:\mathrm{780}\:\mathrm{is}\:\mathrm{wrong}. \\ $$
Answered by Tinkutara last updated on 07/Oct/17
$${Let}\:\mathrm{2}{n}\:{be}\:{the}\:{number}\:{of}\:{sides}\:{of}\:{the} \\ $$$${regular}\:{polygon}.\:{Then}\:{number}\:{of} \\ $$$${scalene}\:{triangles}\:{are}\:{given}\:{by} \\ $$$$\:^{\mathrm{2}{n}} {C}_{\mathrm{3}} \:−\:\mathrm{2}{n}\left(\mathrm{2}{n}\:−\:\mathrm{4}\right)\:−\:\mathrm{2}{n}\left({n}\:−\:\mathrm{1}\right) \\ $$$${So}\:{here}\:{it}\:{gives}\:^{\mathrm{20}} {C}_{\mathrm{3}} −\mathrm{20}×\mathrm{16}−\mathrm{20}×\mathrm{9} \\ $$$$=\mathrm{1140}−\mathrm{320}−\mathrm{180}=\mathrm{640} \\ $$
Commented by Tinkutara last updated on 07/Oct/17
$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{the}\:\mathrm{reason}\:\mathrm{behind}\:\mathrm{this} \\ $$$$\mathrm{formula}.\:\mathrm{Can}\:\mathrm{somebody}\:\mathrm{elaborate} \\ $$$$\mathrm{please}? \\ $$
Commented by alex041103 last updated on 07/Oct/17
$${Ok}.\:{Can}\:{I}\:{explane}\:{it}\:{to}\:{you}\:{by}\:{a}\:{photo}. \\ $$
Commented by Tinkutara last updated on 07/Oct/17
$$\mathrm{Sure}. \\ $$