label vertices of n-gon as A1,A2,A3....,An,
diagonal is a combination of 2 elements from this set ie A1A2 corresponds to a diagonal
hence there are nc2 diagonals, but the n-sides A1A2,A2A3,....are not diagonals, so there are nC2-n diagonals in total
Im not sure about the next problem, most likely wrong, but i'll take a stab...
Consider the separate cases of the maximum numbers of intersections of m-lines, maximum number of intersections of n-circles, and maximum numbers of m-lines with n-circles:
Case 1:
1 line has no intersection with any other line
2 lines has one intersection
3 lines has 1+2 intersections
.......
m-lines have 0+1+2+3+4+......+m=[m(m-1)]/2 intersections
Case 2:
Each circle can have a maximum of 2 intersections with another circle
hence n circles have a maximum of 2[(n-1)+(n-2)+(n-3)+...+2+1]
=2[n(n-1)-[1+2+...+(n-1)]]
=n(n-1)
Case 3:
A line cuts a circle at a maximum of 2 points, hence the maximum number of intersections of m-lines with n-circles is 2mn
Maximum number of points of intersection is: [m(m-1)]/2+n(n-1)+2mn
Most probably wrong......seems a bit iffy imo...