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Home » FRONTLOAD DOPAMINE: Funny Meme Faces You Need to Watch Before Laughter Hits Hard - jntua results

FRONTLOAD DOPAMINE: Funny Meme Faces You Need to Watch Before Laughter Hits Hard - jntua results

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: How many distinct 6-letter arrangements can be formed from the letters of "THEORETICAL" if the letters 'E' and 'A' must not be adjacent?

📌 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!} = 2 \cdot 362880 / 4 = 181440 $. The valid arrangements are $ 302400 - 181440 = 120960 $. Thus, the answer is $ \boxed{120960} $.Question: A 5 cm by 8 cm rectangle is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of $\pi$.

📌 Solution: To find the circumference of the circle in which a rectangle is inscribed, we first recognize that the diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem, the diagonal $d$ of a 5 cm by 8 cm rectangle is:

📌 d = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}

📌 Thus, the diameter of the circle is $\sqrt{89}$ cm, and the radius is $\frac{\sqrt{89}}{2}$ cm. The circumference $C$ of the circle is given by:

📌 C = 2\pi r = 2\pi \cdot \frac{\sqrt{89}}{2} = \pi \sqrt{89}

📌 \boxed{\pi \sqrt{89}}

📌 Question: Compute $\cos 180^\circ + \sin 270^\circ$.

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