Question

Five points are marked on a circle. The number of distinct polygons of three or more sides can be drawn using some (or all) of the five points as vertices is

• A. 10
• B. 12
• C. 14
• D. 16
• E. 18

Hint:

Formula Tip: You can also use the power set! The total number of subsets is $2^5 = 32$. Subtract subsets that don't form polygons: 0 points (1), 1 point (5), and 2 points (10). $32 - 1 - 5 - 10 = 16$.

Answer 1

The Correct Option is D

Concept:
A polygon is formed by selecting $k$ vertices from a given set of $n$ points. The number of ways to choose $k$ points from $n$ points is given by the combination formula ${}^nC_k = \frac{n!}{k!(n-k)!}$. To form a polygon, we need a minimum of 3 points.

Step 1: Identify the possible types of polygons.
We have 5 points available on the circle ($n=5$). A polygon requires 3 or more sides. Therefore, we can form: 1. Triangles (choosing 3 points) 2. Quadrilaterals (choosing 4 points) 3. Pentagons (choosing 5 points)

Step 2: Calculate the number of triangles.
We choose 3 points out of 5 to form triangles: $${}^5C_3 = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{20}{2} = 10$$ There are 10 distinct triangles.

Step 3: Calculate the number of quadrilaterals.
We choose 4 points out of 5 to form quadrilaterals: $${}^5C_4 = \frac{5!}{4!(5-4)!} = \frac{5 \times 4!}{4! \times 1} = 5$$ There are 5 distinct quadrilaterals.

Step 4: Calculate the number of pentagons.
We choose 5 points out of 5 to form pentagons: $${}^5C_5 = 1$$ There is 1 distinct pentagon.

Step 5: Find the total number of polygons.
Add the number of each type of polygon together to get the final answer: $$\text{Total} = 10 + 5 + 1$$ $$\text{Total} = 16$$ Hence the correct answer is (D) 16.