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(C) The $20$ pillars form the vertices of a $20$-sided polygon.Connecting every pair of pillars corresponds to drawing all possible lines (sides and d

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$170$

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(C) The $20$ pillars form the vertices of a $20$-sided polygon.Connecting every pair of pillars corresponds to drawing all possible lines (sides and diagonals) between these $20$ vertices.The total number of ways to choose $2$ pillars out of $20$ is given by the combination formula $^{20}C_2$.$^{20}C_2 = \frac{20 	imes 19}{2} = 190$.These $190$ lines include the $20$ sides of the polygon (which connect adjacent pillars).Since the problem states that beams are connected only to non-adjacent pillars,we must subtract the $20$ sides from the total number of lines.Total number of beams $=$ Total lines $-$ Number of sides $= 190 - 20 = 170$.

Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar is connected by beams to the top of all its non-adjacent pillars,then the total number of beams is:

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