Question

The straight lines l₁, l₂ and l₃ are parallel and lie in the same plane. A total number of m points are taken on l₁, n points on l₂, k points on l₃. The maximum number of triangles formed with vertices at these points are
(a) ^{m + n + k}C₃
(b) ^{m + n + k}C₃ – ^mC₃ – ⁿC₃ – ^kC₃
(c) ^mC₃ + ⁿC₃ + ^kC₃
(d) ^mC₃ × ⁿC₃ × ^kC₃

Answer 1

Here the total number of points are m + n + k .
Which must give ^{m + n + k}C₃ numbers of triangles.
Since m points lie l₁, taking 3 points at a time is ^mC₃.
similarly, from n points on l₂, taking 3 points at a time is ⁿC₃ and ^kC₃ from k,
from these, three points ^mC₃ or ⁿC₃ or ^kC₃ triangle can not be formed
∴ The required number of triangles is ^{m + n + k}C₃ − ^mC₃ − ⁿC₃ − ^kC₃
Hence, the correct answer is option B.