Suppose that 20 pillars of the same height have been erected along the boundary of a 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:

We have a total of $$n = 20$$ identical pillars placed at equal intervals on the circumference of a circular stadium.

Any pair of pillars can potentially be connected by a beam. The number of different pairs that can be chosen from $$n$$ objects is given by the combination formula

$$^nC_2 \;=\; \frac{n(n-1)}{2}.$$

Substituting $$n = 20$$, we get

$$^{20}C_2 \;=\; \frac{20 \times 19}{2} \;=\; 190.$$

These 190 pairs represent all possible connections, including the connections between adjacent pillars (the sides of the 20-gon) and the connections between non-adjacent pillars (the chords of the 20-gon).

However, the problem states that beams are laid only between non-adjacent pillars. We must therefore exclude those pairs which are adjacent.

In a 20-sided polygon, each side joins exactly two adjacent pillars, and there are as many sides as pillars. Hence the number of adjacent pairs equals

$$20.$$

So the number of beams required is obtained by subtracting these 20 adjacent connections from the total 190 possible connections:

$$\text{Required beams} \;=\; 190 \;-\; 20 \;=\; 170.$$

Therefore, the total number of beams that can be fixed between the tops of all non-adjacent pillars is $$170$$. Hence, the correct answer is Option A.