Permutations & Combinations
Line intersection counting
nta_pyq_2025_apr
Grade 12
Question:
Line L of slope 2 and line L of slope 1 2 1 2 intersect at the origin O. In the first quadrant, P, P,$\ldots. P 1 2 12 are 12 points o_n line L and Q, Q,$$\ldots.. Q are 9 points o_n line L. Then the total number of triangles, that can be 1 1 2 9 2 formed having vertices at three of the 22 points O, P, P,$$\ldots P, Q, Q,$$\ldots. Q, is$: 1 2 12 1 2 9
$1080$
$1134$
$1026$
$1188$
Step-by-Step Solution
Key Concept: Use$line-slope$conditions and intersection constraints to count valid selections geometrically.
Total number of $\Delta$ are$(2)$9 12 9 12 1 9$12 = C_{1}$$C_{2} + C_{2}$$C_{1} + C_{1}$$C_{1}$$C_{1} = 594 + 432 + 108 = 1134$
Correct Answer: 2