In a plane, there are 5 straight lines which will pass through a given point, 6 others which all pass through another given point, and 7 others which all as through a third given point. Supposing no three lines intersect at any point and no two are parallel, find the number of triangles formed by the intersection of the straight line.

Explanation

Let 5 straight lines be passing through A, 6 passing through B, and 7 passing through C. In all, there are 18 straight lines.
To find the number of triangles, we have to find the number of selection of 3 lines from these 18 lines, keeping in mind that selection of 3 lines from the lines passing through A, B, or C will not give any triangle.
Hence, the required number of triangles is
`.^(18)C_(3)-(.^(5)C_(3)+ .^(6)C_(3) + .^(7)C_(3))=751`.