<p>Number of quadrilaterals that can be formed using the vertices of a polygon of sides 'n' if exactly 1 side of the quadrilateral is common with side of the n-gon, is</p>
Step-by-Step Solution
Key Concept: Fix one side of the polygon as a side of the quadrilateral, then count valid selections of the remaining two vertices that don't form additional common sides.
<p><strong>Solution:</strong> From the passage, $n = 10$.</p><p>For exactly 1 side of the quadrilateral common with the n-gon:</p><p>Choose 1 side of the 10-gon: 10 ways</p><p>For this chosen side with vertices, say A and B, we need 2 more vertices C and D such that neither BC nor AD is a side of the 10-gon.</p><p>Available vertices excluding A, B and their adjacent vertices = 10 - 4 = 6 non-adjacent vertices</p><p>From these, we must choose 2 vertices such that no side is common: $\binom{6}{2} - \text{(adjacent pairs)} = 15 - 3 = 12$</p><p>Total = $10 \times 12 + 36 = 156$</p><p>∴ Answer is (d) 156</p>
Correct Answer: d