Solution: First, calculate the total number of arrangements of the letters in "THEORETICAL". The word has 10 letters with repetitions: T (2), E (3), H (1), O (1), R (1), I (1), C (1), A (1). Total arrangements: $ \frac{10!}{2! \cdot 3!} = 302400 $. Next, subtract the arrangements where 'E' and 'A' are adjacent. Treat 'EA' or 'AE' as a single entity, reducing the problem to arranging 9 items (with T repeated twice and E reduced to 2). Number of such arrangements: $ 2 \cdot \frac{9!}{2! \cdot 2!}
["Olympiad Geometry Insight: Circumference of the Circle Around an Inscribed Rectangle", "When a rectangle is inscribed in a circle, its diagonal becomes the diameter of the circle. This key geometric property allows us to calculate the circumference using the rectangle’s diagonal length.", "Given a rectangle with dimensions 5 cm by 8 cm:", "- The diagonal forms the hypotenuse of a right triangle with legs equal to the rectangle’s sides.\n- Apply the Pythagorean theorem:\n $$\n d = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}\n $$\n So, the diameter of the circle is $ \sqrt{89} $ cm.", "- The circumference $ C $ of a circle is given by $ C = \pi \ imes \ ext{diameter} $.\n Thus:\n $$\n C = \pi \cdot \sqrt{89}\n $$", "Therefore, the circumference of the circle is $ \boxed{\pi \sqrt{89}} $ cm."]
