Saturday, December 13, 2008

Q) Find the number of triangles that can be formed with the vertexes of a 10-sided-polygon (decagon) as its vertexes ,if
1)The triangles can not have more than one side common with the polygon;
2)The triangle can not have any side common with the polygon .

A) Total triangles possible in general = 10C3
Triangles having more than 1 side common with polygon = Triangles with 2 sides common with polygon = 10 (ten triangles with 2 sides as 2 consecutive sides of the polygon)
1) 10C3 - 10 triangles

2) Triangles with 1 side common with polygon = 10.(6) = 60
(Fix one side, then 8 vertexes are left, and also ignore 1 consecutive vertex on either side, so we are left with 6. And 10 such triangles are possible due to 10 vertexes)
So, triangles with no sides common with polygon = (10C3 - 10) - 60
= 10C3 - 70


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