Step 1 :Let's denote the radius of circle X as \(r_x\) and the radius of circle Y as \(r_y\). The radius of circle C is the sum of the radii of circle X and circle Y, so \(r_c = r_x + r_y\).
Step 2 :The area of circle C is \(\pi (r_x + r_y)^2\).
Step 3 :We can observe that the area of circle C is the sum of the areas of circles X and Y plus the area of the rectangle formed by \(r_x\) and \(r_y\).
Step 4 :The area of the rectangle is \(2 \pi r_x r_y\), and the areas of circles X and Y are \(\pi r_x^2\) and \(\pi r_y^2\) respectively.
Step 5 :Therefore, the area of circle C is \(\pi r_x^2 + \pi r_y^2 + 2 \pi r_x r_y = \pi (r_x + r_y)^2\).
Step 6 :Since the area of circle C is the sum of the areas of circles X and Y plus the area of the rectangle formed by \(r_x\) and \(r_y\), the area of circle C must be greater than the areas of circles X and Y.
Step 7 :Therefore, the area of circle C must be \(1296 \pi\) units \(^{2}\), which is the largest option given in the question.
Step 8 :Final Answer: \(\boxed{1296 \pi}\) units \(^{2}\)