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Home » Add the two equations: 2x = 60 → x = 30. - Tacotoon

Add the two equations: 2x = 60 → x = 30. - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Therefore, the number of arrangements where the two N's are **not adjacent** is:

📌 Thus, the number of valid distinct arrangements is:

📌 \boxed{40}

📌 Question: A smart city integrator in San Francisco is analyzing traffic patterns and observes that 6 autonomous vehicles arrive at an intersection at random times during a 10-minute window. Each vehicle independently chooses a 2-minute interval (possibly overlapping) during the window: [0,10), and we are to compute the probability that all 6 vehicles arrive within some continuous 2-minute subinterval of [0,10), assuming uniform and independent arrival times.

📌 Solution: This is a continuous probability problem involving uniform random variables and coverage intervals.

📌 Let each vehicle arrive at a time uniformly distributed over [0,10). We want the probability that there exists a 2-minute interval [t, t+2) ⊆ [0,10) such that all 6 arrival times fall within [t, t+2).

📌 Standard technique: Compute the complementary probability — that **no** such interval covers all 6 points — and subtract from 1. However, due to complexity, we instead use geometric probability and order statistics.

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Substitute into $ f(\theta) $:
f(\theta) = (1 - 2\sin^2 \theta \cos^2 \theta) - 4\sin^2 \theta \cos^2...
Now use the identity $ \sin(2\theta) = 2\sin\theta \cos\theta $, so:
\sin^2(2\theta) = 4\sin^2\theta \cos^2\theta \Rightarrow \sin^2\theta...
f(\theta) = 1 - 6 \cdot \frac{1}{4} \sin^2(2\theta) = 1 - \frac{3}{2}...
The maximum value of $ \sin^2(2\theta) $ is 1, so the minimum value of...
- \frac{3}{2}(1) = -\frac{1}{2}.
This minimum is achieved when $ \sin^2(2\theta) = 1 $, which occurs fo...
Therefore, the minimum value is $ \boxed{-\frac{1}{2}} $.
Question: A drone specialist analyzing nutrient dispersion in a cloud...

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