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Question Number 21683 by Tinkutara last updated on 30/Sep/17

$$\mathrm{Suppose}{A}_1{A}_2...A_{20}\mathrm{is}\mathrm{a}20-\mathrm{sided}\mathrm{regular}$$$$\mathrm{polygon}.\mathrm{How}\mathrm{many}\mathrm{non}-\mathrm{isosceles}$$$$\left(\mathrm{scalene}\right)\mathrm{triangles}\mathrm{can}\mathrm{be}\mathrm{formed}\mathrm{whose}$$$$\mathrm{vertices}\mathrm{are}\mathrm{among}\mathrm{the}\mathrm{vertices}\mathrm{of}\mathrm{the}$$$$\mathrm{polygon}\mathrm{but}\mathrm{whose}\mathrm{sides}\mathrm{are}\mathrm{not}\mathrm{the}$$$$\mathrm{sides}\mathrm{of}\mathrm{the}\mathrm{polygon}?$$

Commented by alex041103 last updated on 30/Sep/17

$${Isn}^{\prime }tit780?$$

Commented by Tinkutara last updated on 03/Oct/17

$$\mathrm{No}.780\mathrm{is}\mathrm{wrong}.$$

Answered by Tinkutara last updated on 07/Oct/17

$$Let2nbethenumberofsidesofthe$$$$regularpolygon.Thennumberof$$$$scalenetrianglesaregivenby$$$${}^{2n}{C}_3-2n(2n-4)-2n(n-1)$$$$Sohereitgives{}^{20}{C}_3-20\times 16-20\times 9$$$$=1140-320-180=640$$

Commented by Tinkutara last updated on 07/Oct/17

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{the}\mathrm{reason}\mathrm{behind}\mathrm{this}$$$$\mathrm{formula}.\mathrm{Can}\mathrm{somebody}\mathrm{elaborate}$$$$\mathrm{please}?$$

Commented by alex041103 last updated on 07/Oct/17

$$Ok.CanIexplaneittoyoubyaphoto.$$

Commented by Tinkutara last updated on 07/Oct/17

$$\mathrm{Sure}.$$

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