Question

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

• A. 562
• B. 252
• C. 105
• D. 400
• E. 282

Hint:

Any \(k\) vertices chosen from a convex polygon always form a convex \(k\)-gon. Use \(\binom{n}{k}\).

Answer 1

The Correct Option is B

Step 1: Condition.
A pentagon requires 5 vertices.
Step 2: Apply combination.
From 10 vertices, choose any 5: \[ \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]
Step 3: Note.
In a convex polygon, any 5 chosen vertices form a convex pentagon.
Final Answer:
\[ \boxed{252} \]