Solution. As a chord lines are added, it can cut all the chord lines before it; this can be done by choosing 2n points on the circumference, and points 1, ... n are the starting points for the 1st, 2nd .. n'th chort line, and points n+1, ... 2n are the ending points, in the same order. The first line has nothing to intercept, the second intercept the 1st, the 3rd the two preceding ones etc for a total of
p = 0+ 1 + 2 + 3 + ...+ c-1 = c (c - 1) / 2.
As for areas, there is one at first. Every line cuts through at least one area, dividing it in two, increasing the # of areas by one, so c lines add the # of areas by c. If there is an intersection point, the line passes through an additional area, adding the # of areas by one; p intersection points mean p additional areas, for a grand total of 1 + c + p.