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

Q: Wait, why do we use $^{20}C_{2}$? Why not just multiply $20$ by $19$?

Great question! If you multiply $20$ by $19$, you are counting each beam twice—once from pillar A to B, and again from pillar B to A. Since a beam is the same regardless of which end you start from, we use the combination formula $^{n}C_{2} = \frac{n(n-1)}{2}$ to avoid double-counting.

Q: Why do we subtract $20$ at the end? Are those the adjacent pillars?

Exactly! The total number of connections, $^{20}C_{2}$, includes every possible line segment between any two pillars. This includes the lines that form the outer boundary of the stadium (the sides of the $20$-sided polygon). Since the problem specifically asks for non-adjacent connections, we must remove those $20$ boundary lines.

Q: Is there a general formula for this?

Yes, there is! For any $n$ pillars, the total number of non-adjacent connections (which are the diagonals of an $n$-sided polygon) is given by $\frac{n(n-3)}{2}$. If you plug in $n=20$, you get $\frac{20 \times 17}{2} = 170$. It is a beautiful, elegant result!

Question

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 beams is :

Options:

(A) 210

(B) 190

(C) 170

(D) 180

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine standing in the center of a grand, circular stadium. Around you, $20$ majestic pillars rise toward the sky, perfectly spaced along the boundary.

You are tasked with a challenge: to connect these pillars with beams, but with a specific constraint—you can only connect pillars that are not right next to each other. This is a classic puzzle of combinatorics.

### Phase 1: The Total Possibilities

To solve this, we must first think big. Let us ignore the constraint for a moment and consider connecting every single pillar to every other pillar.

Each beam requires two endpoints. Since we have $20$ pillars and we need to choose any $2$ of them to form a beam, this is a combination problem denoted as $^{20}C_{2}$.

The calculation is as follows:

$$^{20}C_{2} = \frac{20 	imes 19}{2 	imes 1} = 190$$

So, there are $190$ possible ways to connect any two pillars in this stadium.

### Phase 2: The Hidden Trap

Now, we must address the constraint. The $190$ connections we just calculated include every possible line segment, including the beams that connect adjacent pillars.

These adjacent connections form the sides of a $20$-sided polygon. Since there are $20$ pillars, there are exactly $20$ such adjacent connections.

These are the beams we are forbidden from using.

### Phase 3: The Elegant Solution

The logic now becomes clear and satisfying. We have the total number of possible connections ($190$) and the number of forbidden, adjacent connections ($20$).

To find the number of beams connecting non-adjacent pillars, we simply subtract the forbidden ones from the total:

$$190 - 20 = 170$$

### Phase 4: The Beauty of Generalization

What we have just discovered is a fundamental property of polygons. The number of non-adjacent connections is equivalent to the number of diagonals in a polygon.

For any $n$-sided polygon, the number of diagonals is given by the formula:

$$\frac{n(n-3)}{2}$$

If we test our result with this formula for $n=20$, we get:

$$\frac{20(20-3)}{2} = \frac{20 	imes 17}{2} = 10 	imes 17 = 170$$

The math aligns perfectly. Whether you think of it as subtracting the boundary from the total or using the diagonal formula, the result is the same: **$170$ beams**.

Select Answer:

Visualized Solution

Visualizing the Stadium

Understanding the Connections

The Total Possible Beams

Combinatorics Logic

Calculating Total Connections

Evaluating Total Connections

The Adjacent Pillars Trap

Counting Adjacent Beams

Subtracting Adjacent Beams

Final Calculation

General Formula (Diagonals of a Polygon)

The Sigma Insight: Combinations and Selection

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Phase 4

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