Line of slope 2 and line of slop intersect at the origin O. In the first quadrant, are 12 points on line and are 9 points on line . Then the total number of triangles, that can be formed having vertices at three of the 22 points O, is : [2

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Solution

Q.
Line $L_{1}$ of slope 2 and line $L_{2}$ of slop $\frac{1}{2}$ intersect at the origin O. In the first quadrant, $P_{1}, P_{2}, . . ., P_{12}$ are 12 points on line $L_{1}$ and $Q_{1}, Q_{2}, . . ., Q_{9}$ are 9 points on line $L_{2}$ . Then the total number of triangles, that can be formed having vertices at three of the 22 points O, $P_{1}, P_{2}, . . ., P_{12}, Q_{1}, Q_{2}, . . ., Q_{9}$ is : [2025]

1 1080

2 1026

3 1134

4 1188

Ans.
(3)

Total number of triangles are

$= {}^{9}C_{1} {}^{12}C_{2} + {}^{9}C_{2} {}^{12}C_{1} + {}^{1}C_{1} \cdot {}^{9}C_{1} {}^{12}C_{1}$

= 594 + 432 + 108 = 1134

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