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

Options

A. 180
B. 210
C. 170
D. 190

Answer

To solve the problem, we need to determine the total number of beams that can be connected between the pillars, excluding the beams that connect adjacent pillars. Here’s the step-by-step solution:

### Step 1: Calculate the total number of ways to connect the pillars
We have 20 pillars, and we want to connect each pillar to every other pillar. The number of ways to choose 2 pillars from 20 to connect them with a beam is given by the combination formula:

\[ \text{Total connections} = \binom{20}{2} \]

Calculating this:

\[ \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190 \]

### Step 2: Identify the number of adjacent pillars
In a circular arrangement, each pillar has exactly 2 adjacent pillars (one on each side). Since there are 20 pillars, the total number of adjacent connections (beams) is:

\[ \text{Adjacent beams} = 20 \]

### Step 3: Calculate the number of non-adjacent beams
To find the number of beams that connect non-adjacent pillars, we subtract the number of adjacent beams from the total connections:

\[ \text{Non-adjacent beams} = \text{Total connections} - \text{Adjacent beams} \]

Substituting the values we calculated:

\[ \text{Non-adjacent beams} = 190 - 20 = 170 \]

### Conclusion
The total number of beams connecting non-adjacent pillars is **170**.

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