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 :

Question

A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?

Answer 1

To determine the total number of beams, we need to understand the scenario: We have 20 pillars in a circular arrangement. Each pillar is connected to all its non-adjacent pillars by a beam.

First, calculate the total possible connections (or edges) between 20 pillars. Each pillar is a vertex in a polygon, and the goal is to connect every pair of vertices.
This is calculated using combinations. For any $n$ vertices in a polygon, the total number of connections or line segments that can be drawn is given by: C(n, 2) = \frac{n(n-1)}{2}.
Substituting $n = 20$, we get: C(20, 2) = \frac{20 \times 19}{2} = 190.
However, connections to adjacent pillars do not count. For each pillar, there are exactly 2 adjacent pillars that should not be connected directly. Thus, 20 pillars will have a total of 20 such adjacent edge connections.
Hence, the number of beams connecting non-adjacent pillars is: 190 - 20 = 170.

Therefore, the total number of beams is 170, which matches the given correct answer.

Answer 2

To determine the total number of beams, we need to understand the scenario: We have 20 pillars in a circular arrangement. Each pillar is connected to all its non-adjacent pillars by a beam.

First, calculate the total possible connections (or edges) between 20 pillars. Each pillar is a vertex in a polygon, and the goal is to connect every pair of vertices.
This is calculated using combinations. For any $n$ vertices in a polygon, the total number of connections or line segments that can be drawn is given by: C(n, 2) = \frac{n(n-1)}{2}.
Substituting $n = 20$, we get: C(20, 2) = \frac{20 \times 19}{2} = 190.
However, connections to adjacent pillars do not count. For each pillar, there are exactly 2 adjacent pillars that should not be connected directly. Thus, 20 pillars will have a total of 20 such adjacent edge connections.
Hence, the number of beams connecting non-adjacent pillars is: 190 - 20 = 170.

Therefore, the total number of beams is 170, which matches the given correct answer.

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 :

The Correct Option is C

Solution and Explanation

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