On a coordinate plane, 12 points are arranged as shown in the figure to the right. If we are to select three points as vertices of a triangle, how many triangles are possible in total? First, there are $\mathbf { K L M }$ ways to select three points from the 12 points. Next, let us find how many ways it is possible to select three or more points in a straight line. Let us look at the two cases. (i) There are $\mathbf { N }$ straight lines that pass through four points. (ii) There are $\mathbf { O }$ straight lines that pass through three points. Hence, among all combinations of three points that are in a straight line and so cannot be the vertices of a triangle, $\mathbf { P Q }$ combinations belong to case (i), and $\mathbf { Q }$ combinations belong to case (ii). Thus, the total number of possible triangles is $\mathbf{STU}$. In particular, if we set $( 1,1 )$ as point A and $( 4,1 )$ as point B , then $\mathbf { V W }$ triangles have two vertices on segment AB.
On a coordinate plane, 12 points are arranged as shown in the figure to the right. If we are to select three points as vertices of a triangle, how many triangles are possible in total?
First, there are $\mathbf { K L M }$ ways to select three points from the 12 points.
Next, let us find how many ways it is possible to select three or more points in a straight line.
Let us look at the two cases.\\
(i) There are $\mathbf { N }$ straight lines that pass through four points.\\
(ii) There are $\mathbf { O }$ straight lines that pass through three points.
Hence, among all combinations of three points that are in a straight line and so cannot be the vertices of a triangle, $\mathbf { P Q }$ combinations belong to case (i), and $\mathbf { Q }$ combinations belong to case (ii).
Thus, the total number of possible triangles is $\mathbf{STU}$.
In particular, if we set $( 1,1 )$ as point A and $( 4,1 )$ as point B , then $\mathbf { V W }$ triangles have two vertices on segment AB.