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Home » Multiply both sides by \( a \): - DNSFLEX

Multiply both sides by \( a \): - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 A cartographer maps a rectangular region that is 150 km wide and 240 km long. If each grid square on the map represents 6 square kilometers, how many grid squares are needed to cover the entire region?

📌 Area of region = 150 × 240 = <<150*240=36000>>36,000 km².

📌 Number of grid squares = 36,000 ÷ 6 = <<36000/6=6000>>6,000.

📌 The retired scientist helps students publish papers. In one year, one student publishes 2 papers, and each paper is co-authored with 3 others, including the mentor. How many unique students are involved in all publications, including the scientist?

📌 Each paper involves 4 individuals: student + 3 co-authors.

📌 But the mentor is always present and shared.

📌 For 2 papers: 2 × 3 = 6 co-authorship slots. Since mentor appears in all, at least 1 other per paper.

📌 Assuming no repeats, maximum unique co-authors = 6 slots, but if the mentor is shared, minimal unique = 1 mentor + 6 = 7, but likely fewer due to overlap.

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Better idea: Use **inclusion-exclusion over the number of adjacent pai...
Instead, we use a **constructive endangered sequence method**:
We attempt to **count valid sequences by placing high-frequency letter...
Let’s suppose we place the **C’s** with no two adjacent.
To place 3 C’s with no two adjacent in 8 positions: we first place 3 C...
The number of ways to choose 3 non-adjacent positions in 8 is a classi...
To place 3 non-consecutive positions: let the positions be $ x_1 < x_2...
For each such choice of 3 non-adjacent positions, we place 3 non-adjac...
Now, from the remaining 5 positions, we must place 2 A’s and 3 G’s (si...
But A and G are different, so A and G can be adjacent.

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