Question

There are n points in a plane, no three of which are collinear except ‘p’ points all of which are on a line. How many 

(a)  straight lines can be formed and 

(b)  triangles can be formed out of these n points?

Answer 1

(a) To form a line we need two points; and these two points may be chosen in ⁿC₂ ways; but, it happens that ‘p’ of the ‘n’ points are on a line; consequently, these points would form only one line instead of  ^pC₂.

Therefore, number of lines =  ⁿC₂ - ^pC₂ + 1

(b)  Number of triangles =  ⁿC₃ - ^pC₃