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Suppose-A-1-A-2-A-20-is-a-20-sided-regular-polygon-How-many-non-isosceles-scalene-triangles-can-be-formed-whose-vertices-are-among-the-vertices-of-the-polygon-but-whose-sides-are-not-the-side




Question Number 21683 by Tinkutara last updated on 30/Sep/17
Suppose A_1 A_2 ...A_(20)  is a 20-sided regular  polygon. How many non-isosceles  (scalene) triangles can be formed whose  vertices are among the vertices of the  polygon but whose sides are not the  sides of the polygon?
SupposeA1A2A20isa20sidedregularpolygon.Howmanynonisosceles(scalene)trianglescanbeformedwhoseverticesareamongtheverticesofthepolygonbutwhosesidesarenotthesidesofthepolygon?
Commented by alex041103 last updated on 30/Sep/17
Isn′t it 780?
Isntit780?
Commented by Tinkutara last updated on 03/Oct/17
No. 780 is wrong.
No.780iswrong.
Answered by Tinkutara last updated on 07/Oct/17
Let 2n be the number of sides of the  regular polygon. Then number of  scalene triangles are given by  ^(2n) C_3  − 2n(2n − 4) − 2n(n − 1)  So here it gives^(20) C_3 −20×16−20×9  =1140−320−180=640
Let2nbethenumberofsidesoftheregularpolygon.Thennumberofscalenetrianglesaregivenby2nC32n(2n4)2n(n1)Sohereitgives20C320×1620×9=1140320180=640
Commented by Tinkutara last updated on 07/Oct/17
I don′t know the reason behind this  formula. Can somebody elaborate  please?
Idontknowthereasonbehindthisformula.Cansomebodyelaborateplease?
Commented by alex041103 last updated on 07/Oct/17
Ok. Can I explane it to you by a photo.
Ok.CanIexplaneittoyoubyaphoto.
Commented by Tinkutara last updated on 07/Oct/17
Sure.
Sure.

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