The first vertex can be selected in 10 ways. The remaining two are to be

Number of triangles using the vertices of a regular decagon:

A regular decagon has 10 vertices. We need to find the number of triangles that can be formed using these vertices, without using the sides of the decagon.

Approach:

To solve this problem, we can use combinations and permutations.

Step 1: Selecting the vertices:

To form a triangle, we need to select 3 vertices from the 10 vertices of the decagon.

The number of ways to select 3 vertices from a set of 10 is given by the combination formula:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of elements in the set and r is the number of elements to be selected.

In this case, n = 10 and r = 3.

C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120

So, there are 120 ways to select 3 vertices from the decagon.

Step 2: Eliminating triangles formed by connecting adjacent vertices:

Since we are not allowed to use the sides of the decagon to form triangles, we need to eliminate the triangles formed by connecting adjacent vertices.

A regular decagon has 10 vertices, and each vertex is adjacent to two other vertices. So, we need to eliminate 2 triangles for each vertex, resulting in a total of 20 eliminated triangles.

Step 3: Calculating the final number of triangles:

The final number of triangles can be calculated by subtracting the eliminated triangles from the total number of ways to select 3 vertices.

Final number of triangles = Total number of ways to select 3 vertices - Eliminated triangles
= 120 - 20
= 100

Therefore, the correct answer is option 'B' - 50 triangles can be formed using the vertices of a regular decagon, without using the sides.