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Home » \( 36 + 3 = 39 \) - Dyverse

\( 36 + 3 = 39 \) - Dyverse

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Therefore, the number of ways to arrange the flowers such that each type appears at least once is \(\boxed{6}\).

📌 Question:** A community health discussion involves 7 participants, including 4 health professionals and 3 community members. In how many ways can they sit around a circular table if all participants are distinguishable?

📌 For circular arrangements, the formula is \((n-1)!\) because one position is fixed to avoid identical rotations. Here, \( n = 7 \).

📌 Thus, the number of circular permutations is:

📌 Calculating \( 6! \):

📌 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

📌 Therefore, the number of ways to arrange the participants around the table is \(\boxed{720}\).Question: An equilateral triangle-shaped solar panel array has an area of $ 36\sqrt{3} \, \text{cm}^2 $. If each side is reduced by 4 cm, by how many square centimeters does the area decrease?

📌 Solution: The area of an equilateral triangle is given by $ \frac{\sqrt{3}}{4} s^2 $, where $ s $ is the side length. Setting $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $, we solve for $ s^2 = 144 $, so $ s = 12 \, \text{cm} $. The new side length is $ 12 - 4 = 8 \, \text{cm} $. The new area is $ \frac{\sqrt{3}}{4} \times 8^2 = 16\sqrt{3} \, \text{cm}^2 $. The decrease in area is $ 36\sqrt{3} - 16\sqrt{3} = 20\sqrt{3} \, \text{cm}^2 $. \boxed{20\sqrt{3}}

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List all prime numbers less than 50:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
There are 15 primes in total, of which 14 are odd and 1 (2) is even.
Total number of ways to choose 3 distinct primes:
inom{15}{3} = rac{15 \cdot 14 \cdot 13}{6} = 455
Number of favorable outcomes (include 2 and two odd primes):
Choose 2 from the 14 odd primes:
inom{14}{2} = rac{14 \cdot 13}{2} = 91
rac{91}{455} = rac{1}{5}
So, the probability that the sum is even is $oxed{\dfrac{1}{5}}$.

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