1. The problem asks to evaluate permutation expressions and find the number of unique permutations of letters in given sets. 2. Recall that permutation $nP r = \frac{n!}{(n-r)!}$. 3. For question 3, symbols Θ, ☀, ♥, ▲ taken two at a time is $4P2 = \frac{4!}{(4-2)!} = \frac{4!}{2!} = \frac{24}{2} = 12$. 4. For question 4, letters A, B, C, D taken all at once or two at a time wasn't explicitly asked; we only have the set without operation. Assuming asked similarly for permutations taken two at a time: $4P2 = 12$. 5. Evaluate each permutation: - 5) $4P2 = \frac{4!}{2!} = 12$ - 6) $5P1 = \frac{5!}{4!} = 5$ - 7) $8P2 = \frac{8!}{6!} = 8 \times 7 = 56$ - 8) $6P6 = \frac{6!}{0!} = 6! = 720$ - 9) $4 + 7P4 = 4 + \frac{7!}{3!} = 4 + \frac{5040}{6} = 4 + 840 = 844$ - 10) $5 \cdot 6P5 = 5 \times \frac{6!}{1!} = 5 \times 720 = 3600$ 6. For number of unique permutations of letters in each word, if words were given they would be calculated as $n!$ where $n$ is the number of letters, assuming no repeating letters. Final answers: - 3) 12 - 4) 12 - 5) 12 - 6) 5 - 7) 56 - 8) 720 - 9) 844 -10) 3600