There are 20 pillars of equal height on a circular ground any two non adjacent pillars joined by a beam. Then the number of such beams are:

Options

A. 180
B. 210
C. `.^(20)C_(2)-20`
D. `.^(20)C_(2)`

Answer

To solve the problem of finding the number of beams that can be formed by connecting non-adjacent pillars on a circular ground with 20 pillars, we can follow these steps:

### Step 1: Understand the problem
We have 20 pillars arranged in a circle. We need to connect any two non-adjacent pillars with beams.

### Step 2: Calculate the total number of ways to choose 2 pillars
The total number of ways to choose 2 pillars from 20 is given by the combination formula \( C(n, r) \), where \( n \) is the total number of pillars and \( r \) is the number of pillars to choose. Here, \( n = 20 \) and \( r = 2 \).

\[ C(20, 2) = \frac{20 \times 19}{2} = 190 \]

### Step 3: Calculate the number of adjacent beams
In a circular arrangement of 20 pillars, each pillar has exactly 2 adjacent pillars. Therefore, the number of adjacent beams (connections between adjacent pillars) is equal to the number of pillars, which is 20.

### Step 4: Subtract the adjacent beams from the total beams
To find the number of non-adjacent beams, we subtract the number of adjacent beams from the total number of beams.

\[ \text{Number of non-adjacent beams} = C(20, 2) - \text{Number of adjacent beams} \]
\[ = 190 - 20 = 170 \]

### Conclusion
The number of beams that can be formed by connecting non-adjacent pillars is 170.

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