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Home » a^4 + b^4 = 1^2 - 2 \cdot \frac14 = 1 - \frac12 = \frac12 - jntua results

a^4 + b^4 = 1^2 - 2 \cdot \frac14 = 1 - \frac12 = \frac12 - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 64000 ÷ 2¹ = 32000

📌 Find smallest n such that 64000 / 2^n = 1 → 2^n = 64000 → n = log₂(64000)

📌 But 2¹⁵ = 32768, 2¹⁶ = 65536

📌 > 32768, < 65536 → so 16 splits to reach ≤1? But we want exactly 1.

📌 However, 64000 is not a power of 2, so we cannot reach exactly 1 via repeated halving in integer division unless we allow floating.

📌 But in practice, we split until each group is 1 — so must go until size 1. So:

📌 Start: 64000

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Solution: This is a binomial probability problem with $ n = 4 $ trials...
The binomial probability formula is:
P(k) = \binom{n}{k} p^k (1-p)^{n-k}
Substitute values:
P(2) = \binom{4}{2} \left(\frac{1}{6}\right)^2 \left(\frac{5}{6}\right...
\binom{4}{2} = 6, \quad \left(\frac{1}{6}\right)^2 = \frac{1}{36}, \qu...
So:
P(2) = 6 \times \frac{1}{36} \times \frac{25}{36} = \frac{150}{1296} =...
Thus, the probability is $\boxed{\dfrac{25}{216}}$
Question: How many distinct words can be formed using all the letters...

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