Question

No. of triangles that can be formed using the vertices of a polygon of 10 sides as their vertices and having exactly one side common with the polygon is

Answer 1

Total number of triangles formed by joining the vertices of n-sided polygon N=number of ways of selecting 3 vertices out of n=(n3)N=n(n−1)(n−2)6∀  n≥3

Number of triangles having one side common with that of the polygon N1=(n−4)nNumber of triangles having two sides common with that of the polygon N2=nIf N0 is the number of triangles having no side common with that of the polygon then we have N=N0+N1+N2N0=N−N1−N2=(n3)−(n−4)n−n=n(n−1)(n−2)6−n2+3nN0=n(n−4)(n−5)6The above formula (N0) is valid for polygon having n no. of the sides such that   n≥6

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