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Chapter 5: Problem 409

The vertices of a regular polygon of 12 sides are joined to form triangles. The number of triangles which do not have their sides as the sides of the polygon is (a) 96 (b) 108 (c) 112 (d) 220

Short Answer

Expert verified
The number of triangles which do not have their sides as the sides of the polygon is 208. However, this answer does not match any of the given options. Please verify the options provided.

Step by step solution

01

Find the total number of triangles that can be formed using vertices of the polygon

A polygon with 12 sides has 12 vertices. In order to form a triangle, we need to select 3 vertices from the 12 available vertices. This can be done using combinations, which can be calculated as follows: Total number of triangles = \(\begin{pmatrix}12 \\ 3\end{pmatrix} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!}\)
02

Calculate the value of the total number of triangles

By performing the calculations, we can find the value of \(\begin{pmatrix}12 \\ 3\end{pmatrix}\) as: Total number of triangles = \(\frac{12!}{3!9!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 2 \times 11 \times 10 = 220\)
03

Find the number of triangles which have their sides as the sides of the polygon

Now we need to find the number of triangles whose sides are also the sides of the polygon. These triangles are formed by selecting 3 consecutive vertices of the polygon. Since the polygon has 12 sides and is regular, it has rotational symmetry. This means that there will be 12 triangles with their sides as the sides of the polygon (one triangle for each vertex, as we go around the polygon).
04

Calculate the number of triangles which do not have their sides as the sides of the polygon

Now we have the total number of triangles and the number of triangles with their sides as the sides of the polygon. We can calculate the number of triangles which do not have their sides as the sides of the polygon by subtracting the latter from the former: Number of triangles which do not have their sides as the sides of the polygon = Total number of triangles - Number of triangles with their sides as the sides of the polygon = 220 - 12 = 208 From the given options, (a) 96 (b) 108 (c) 112 (d) 220 None of them matches the calculated answer, which is 208. Please ensure that the options provided in the exercise are correct.

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