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boy returns lost ring to owner - RTA

📅 May 9, 2026 👤 scraface

May 09, 2026

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📌 \binom{5}{2} = \frac{5!}{2!3!} = \frac{5 \times 4}{2} = 10

📌 \binom{6}{2} = \frac{6!}{2!4!} = \frac{6 \times 5}{2} = 15

📌 \text{Total panels} = 10 \times 15 = 150

📌 So, 150 distinct panels can be formed.

📌 An urban forestry researcher photographs trees in a grid of 4 rows and 3 columns. If she selects 3 non-overlapping 1×1 grids (positions) at random, what is the probability that all 3 are in different rows?

📌 Total number of 1×1 grid positions: \( 4 \times 3 = 12 \).

📌 Number of ways to choose 3 distinct positions: \( \binom{12}{3} = \frac{12 \times 11 \times 10}{6} = 220 \).

📌 Now, favorable outcomes: all 3 positions in different rows.

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Question: Find the maximum value of \((\sin x + \cos x)^2 + ( an x + \...
Solution: First, expand:
Let \(u = \sin^2 x\), then \(\cos^2 x = 1 - u\), so:
rac{u}{1 - u} + rac{1 - u}{u} + 2 = rac{u^2 + (1 - u)^2}{u(1 - u)} +...
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an x + \cot x = rac{\sin x}{\cos x} + rac{\cos x}{\sin x} = rac{\si...
Now the full expression:
+ \sin 2x + rac{4}{\sin^2 2x}
Let \(t = \sin 2x\), \(0 < t \leq 1\) (since \(x \in (0, \pi/2)\Righta...
Define \(f(t) = 1 + t + rac{4}{t^2}\), \(0 < t \leq 1\)

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