Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ items from a set of $n$ items, where order doesn't matter, is given by: $^nC_r = \frac{n!}{r!(n-r)!}$
• Polygons: A polygon with $n$ vertices has $n$ sides.
• Diagonals of a polygon: The number of diagonals in an $n$-sided polygon is given by $^nC_2 - n$.

Step-by-Step Solution

Step 1: Calculate the Total Number of Possible Connections

• Explanation: We want to find the total number of ways to connect any two pillars without any restrictions. Since the order of selection doesn't matter (connecting pillar A to pillar B is the same as connecting pillar B to pillar A), we use combinations. We have 20 pillars and we want to choose 2 to form a connection.
• Calculation: Using the combination formula with $n=20$ and $r=2$: $^{20}C_2 = \frac{20!}{2!(20-2)!} = \frac{20!}{2!18!} = \frac{20 \times 19}{2 \times 1} = 10 \times 19 = 190$
• Reasoning: This result, 190, represents the total number of ways to choose any two pillars out of the 20, irrespective of whether they are adjacent or not.

Step 2: Identify and Subtract the Excluded Connections (Adjacent Pillars)

• Explanation: The problem specifies that we only want to connect non-adjacent pillars. Adjacent pillars are those that are next to each other along the circular boundary. Connecting adjacent pillars would form the sides of a 20-sided polygon. We need to subtract these connections from the total number of possible connections.
• Calculation: A 20-sided polygon has 20 sides. Therefore, there are 20 connections between adjacent pillars. $\text{Number of adjacent connections} = 20$
• Reasoning: Each pillar has two adjacent pillars. Connecting each pillar to its two adjacent pillars would create the 20 sides of the polygon.

Step 3: Calculate the Total Number of Beams (Non-Adjacent Connections)

• Explanation: To find the number of beams connecting only non-adjacent pillars, we subtract the number of adjacent connections (sides of the polygon) from the total number of possible connections.
• Calculation: $\text{Total number of beams} = (\text{Total possible connections}) - (\text{Number of adjacent connections}) = 190 - 20 = 170$
• Reasoning: By subtracting the connections between adjacent pillars, we are left with only the connections between non-adjacent pillars, which are the beams we want to count.

Common Mistakes & Tips

• Confusion between Combinations and Permutations: Remember to use combinations when the order of selection doesn't matter.
• Misunderstanding "Non-adjacent": "Non-adjacent" means excluding the connections that form the sides of the polygon.
• Using the Formula Directly: Remembering the formula for the number of diagonals in a polygon ($^nC_2 - n$) can save time.

Summary

The problem asks us to find the number of beams connecting non-adjacent pillars among 20 pillars arranged in a circle. We first calculated the total number of ways to connect any two pillars, which is $^{20}C_2 = 190$. Then, we subtracted the number of connections between adjacent pillars, which is 20 (the number of sides of the 20-sided polygon). The result is $190 - 20 = 170$.

The final answer is $\boxed{170}$, which corresponds to option (C).