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Home » \( \int 4x^3 \, dx = x^4 \), - Aurero

\( \int 4x^3 \, dx = x^4 \), - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Alternatively: number of ways to partition 4 labeled items into two unlabeled pairs:

📌 \frac{1}{2} \binom{4}{2} = 3

📌 \(2,1,1\) → (2,1,1): One archive has 2, two have 1

📌 Number of ways: Choose the pair: \(\binom{4}{2} = 6\), then assign the remaining two to separate archives. Since the two singles are indistinguishable in size, and archives are indistinguishable, no overcounting — 6 ways.

📌 Total: \(1 + 4 + 3 + 6 = 14\)

📌 Thus, the number of distinct assignments is \(\boxed{14}\).

📌 A glaciologist models daily melt rates using a fair six-sided die to simulate random variability: each roll determines the melt level (1–6 units). If she rolls the die once per day for 5 consecutive days, what is the probability that she observes exactly two distinct melt values?

📌 We compute the probability that in 5 rolls of a fair six-sided die, exactly two distinct numbers appear.

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