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

Step-by-step solution

Each of the beams connects the top of two non-adjacent pillars, hence we have to find in how many ways we can select the required two pillars which are non-adjacent out of the given 20 pillars. As, C 2   20 is the number of beams connecting any two pillars whereas 20 is the number of adjacent beams. Thus, the total number of beams = C 2   20 - 20 Using C r n = n ! r ! n - r ! , = 20 ! 2 ! × 18 ! - 20 = 20 × 19 × 18 ! 2 ! × 18 ! - 20 = 190 - 20 = 170 .