Great solution! To understand why these formulas work, I think it is easiest to solve the problem graphically. Draw a line from (0, c) to (c, 0). This is your boundary of x + y <= c. Now, on top of this you are going to draw various rectangles to represent the two uniform variables [0, a], [0, b] along each axis. Take the first case of a > c and b > c. Here you have a rectangle ab with a triangle in the lower left corner from (0, c) to (c, 0). You only want this triangle in the lower left corner since these are your allowed values of x + y <= c. The area of this triangle = 1/2 * base * height = 0.5 * c^2. The area of the rectangle is ab. The odds of landing in the "allowed" triangle are therefore 0.5 * c^2 / ab. You need to rationalize this so that the numerator and denominator are integers, so c^2 / 2ab. This gives the first case "if a >= c and b >= c" in the logic above. The rest are similar, draw pictures to find the "allowed" region within the square.