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baker v carr summary - RedTag Digital

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Solution: This is equivalent to partitioning 4 distinguishable objects into 2 non-empty indistinct subsets, calculated by the Stirling number $ S(4, 2) $. Using the recurrence:

📌 S(4, 2) = S(3, 1) + 2 \times S(3, 2) = 1 + 2 \times 3 = 7

📌 Thus, there are $ \boxed{7} $ valid distributions.Question: The average of $3u - 4$, $7u + 2$, and $4u - 1$ is ... If $u$ is a positive multiple of 3 and $u^2$ is less than 100, what is the average?

📌 Solution: First, calculate the sum of the expressions: $(3u - 4) + (7u + 2) + (4u - 1) = 14u - 3$. Divide by 3 to find the average: $\frac{14u - 3}{3}$. Since $u$ is a positive multiple of 3 and $u^2 < 100$, possible values for $u$ are 3, 6. Testing $u = 3$: $\frac{14(3) - 3}{3} = \frac{42 - 3}{3} = \frac{39}{3} = 13$. For $u = 6$, $u^2 = 36 < 100$, but $14(6) - 3 = 81$, $\frac{81}{3} = 27$. However, the problem implies a unique answer, so the smallest valid $u = 3$ gives $\boxed{13}$.

📌 Question: An AI development specialist allocates $2 \frac{1}{2}$ terabytes of storage for training models. If $ \frac{3}{4} $ terabytes are used for neural network weights, what fraction of the total storage remains?

📌 Solution: Convert $2 \frac{1}{2}$ to an improper fraction: $\frac{5}{2}$. Subtract $\frac{3}{4}$: $\frac{5}{2} - \frac{3}{4} = \frac{10}{4} - \frac{3}{4} = \frac{7}{4}$. The remaining storage is $\frac{7}{4}$ terabytes, which is $\boxed{\dfrac{7}{4}}$ of the total.

📌 Question: A computational biologist measures two protein chains at 3.5 inches and 8.5 inches. What is the average length, in inches, of the chains?

📌 Solution: Add the lengths: $3.5 + 8.5 = 12$. Divide by 2: $\frac{12}{2} = 6$. The average length is $\boxed{6}$ inches.

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