by using the vertices of a regular polygon of 12 sides, Pentagons are formed. Then the number of Pentagons having no side common with the given polygon.

To find the number of pentagons that can be formed using the vertices of a regular dodecagon (12-sided polygon) without sharing any sides with the dodecagon, we can follow a systematic approach.

Understanding the Problem

A regular dodecagon has 12 vertices. We want to select 5 of these vertices to form a pentagon, ensuring that no two selected vertices are adjacent, as that would create a common side with the dodecagon.

Choosing Vertices

When selecting 5 vertices from the 12, we need to ensure that there is at least one vertex between any two chosen vertices. This can be visualized as placing "gaps" between the selected vertices.

Using Combinatorial Methods

To solve this, we can use a combinatorial approach. First, we can think of the 12 vertices as arranged in a circle. If we select 5 vertices, we will have 7 vertices left, which we can think of as gaps.

Calculating the Gaps

• We need to place 5 gaps between the selected vertices.
• This leaves us with 2 additional gaps that can be placed anywhere among the 5 chosen vertices.

Applying the Stars and Bars Theorem

Using the stars and bars method, we can distribute these 2 additional gaps among the 5 chosen vertices. The formula for this is:

Number of ways = C(n + k - 1, k - 1)

Where n is the number of gaps (2) and k is the number of selected vertices (5).

Final Calculation

Thus, we need to calculate:

C(2 + 5 - 1, 5 - 1) = C(6, 4) = 15

Result

Therefore, the total number of pentagons that can be formed using the vertices of a regular dodecagon, without sharing any sides, is 15.