How many triangles and quadrilaterals altogether can be formed using the vertices of a 7-sided regular polygon?

A 35

B 40

C 50

D 65

E 70

## How many triangles and quadrilaterals altogether can be formed using the vertices of a 7-sided regular polygon?

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• Number of triangles can be formed from a polygon with 7 sides = 7C3 = (7*6*5) / (1*2*3) = 35;

• Number of quadrilaterals can be formed from a polygon with 7 sides = 7C4 = (7*6*5*4) / (1*2*3*4) = 35;

Total Number of ways that triangles and rectangles can be formed from a polygon with 7 sides = 35 + 35 = 70.

Correct answer: E

Hope this helps!

-Jay

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**Solution:**

There are 7C3 = (7 x 6 x 5) / (3 x 2) = 35 ways to choose 3 vertices from the 7 vertices of the polygon to form a triangle. Likewise, there are 7C4 = (7 x 6 x 5 x 4)/(4 x 3 x 2) = 35 ways to choose 4 vertices from the 7 vertices of the polygon to form a quadrilateral. Therefore, there is a total of 35 + 35 = 70 triangles and quadrilaterals that can be formed.

**Answer: E**

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