1) When A is excluded

We need to select 2 points from 1 line and 1 from the other.

Total number of ways

= mC2*nC1 + nC2*mC1

\(= \frac{m(m-1)}{2} * n+ \frac{n(n-1)}{2} *m\)

\(= \frac{mn}{2} * (m-1+n-1)\)

\(= \frac{mn}{2} * (m+n-2)\)

2) When A is included

In addition to the triangles we can form in the above case, we can form triangle that must include point A.

Total number of triangles that can be form, when A must be included = mC1*nC1 = mn

Total number of triangles possible in this case

\(= \frac{mn}{2} * (m-1+n-1) + mn\)

\(=\frac{mn}{2} * (m-1+n-1+2)\)

\(= \frac{mn}{2} * (m+n)\)


The ration of number of triangles in the two cases is: \(= \frac{m+n-2}{m+n}\)