Question

How many heptagons can be drawn by joining the vertices of a polygon with 10 sides?

• A. 562
• B. 120
• C. 105
• D. 400
• E. 282

Hint:

For a convex \(n\)-gon, any \(k\) chosen vertices form a convex \(k\)-gon. Use \(\binom{n}{k}\).

Answer 1

The Correct Option is B

Step 1: Condition.
A heptagon requires 7 vertices. Out of 10 vertices, we must choose 7.

Step 2: Apply combination formula.
\[ \binom{10}{7} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
Step 3: Note.
In a convex polygon, any 7 chosen vertices form a convex heptagon.
Final Answer:
\[ \boxed{120} \]