1. Interpret the arrangement

There are $20$ pillars placed on the boundary of a circular stadium, so their tops form the vertices of a $20$-gon.

A beam is drawn from each pillar to the tops of all its non-adjacent pillars.

That means:

• A pillar is not connected to itself.
• A pillar is not connected to its two adjacent pillars.
• It is connected to all the remaining pillars.

2. Number of non-adjacent pillars for one pillar

For any one pillar:

• Total other pillars $= 20 - 1 = 19$
• Adjacent pillars $= 2$

So, non-adjacent pillars connected to it: $19 - 2 = 17$

Thus each pillar appears to have $17$ beams from it.

3. Count total connections carefully

If we multiply directly: $20 \times 17 = 340$

But this counts every beam twice:

• once from pillar $A$ to pillar $B$
• once from pillar $B$ to pillar $A$

Therefore total number of distinct beams is: $\frac{20 \times 17}{2} = 170$

4. Alternative check using diagonals of a polygon

Connecting each vertex of a $20$-gon to all non-adjacent vertices gives exactly the number of diagonals of the polygon.

Number of diagonals in an $n$-gon is: $\frac{n(n-3)}{2}$

For $n=20$: $\frac{20(20-3)}{2} = \frac{20\cdot 17}{2} = 170$

5. Check options

• A: $180$ ❌
• B: $210$ ❌
• C: $170$ ✅
• D: $190$ ❌

So the correct option is C.