# The number of triangles and lines

Consider 8 vertices of a regular octagon and its centre. If T denotes the number of triangles and S denotes the number of straight lines that can be formed with these 9 points then T - S has the value a) 44 b) 48 c) 52 d) 56

• What have you done? – Apurv Oct 22 '15 at 8:44
• T = 9C3 -4=80 , S = 9C2 - 8 = 28 , so T - S = 80 - 28 = 52 – Ravi Oct 22 '15 at 8:49

The solution you have given as a comment is correct: Any three of the nine given points determine a "triangle". Four of these triangles are degenerate, and should not be counted. Any two vertices of the octagon determine a line. In this way the possible lines through the center of the octagon are already counted. It follows that $$T-S={9\choose 3}-4+{8\choose 2}=52\ .$$