Permutations & Combinations
Polygon triangle counting
nta_pyq_2025_apr
Grade 12
Question:
Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If$p + q = 126$, then the 2 2 y eccentricity of the ellipse is: x + = 1 16 n
$3 4$
$1 2 \sqrt{7}$
$4$
$1 \sqrt{2}$
Step-by-Step Solution
Key Concept: Count polygon triangles and exclude or include special configurations using combinations of vertices.
h Total trangles =$\Rightarrow$=$C_{3}$h$(4)$Total$auadrilaterals =$$C_{4}$$= q$n n$n+1$$C_{3} +$$C_{4}$$= 126$$\Rightarrow$$C_{4} = 126$$\Rightarrow$$n + 1 = 9$$\Rightarrow$$n = 8$2 2 2 2 x y x y + = 1 $\Rightarrow$ + = 1 16 n 16 8 8 8 1 e =$\sqrt1 - =$$\$sqrt = 16$16$$\sqrt{2}$
Correct Answer: 4