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Home » Solution: We recognize that this is a sum of cubes pattern: - ClickBalance

Solution: We recognize that this is a sum of cubes pattern: - ClickBalance

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Question: A soil scientist distributes 6 identical nutrient samples into 4 distinct soil test chambers. How many ways can this be done if each chamber must receive at least one sample?

📌 Question: A bioinformatics engineer sequences 5 genes, each with 2 possible alleles. How many distinct allele combinations exist if exactly 3 genes must have the first allele and 2 must have the second?

📌 Solution: This is a multinomial coefficient. The number of ways is $

📌 Let us factor this expression. Group terms as follows:

📌 The root on the positive imaginary axis is:

📌 Note that all roots lie on or inside the unit circle because $P(z)$ is palindromic with positive coefficients, and no root lies exactly on the unit circle with non-zero real part unless its a root of unity, which is not the case here.

📌 Check $f(-1) = -1 -1 -1 -1 + 1 = -3$, $f(0) = 1$, $f(1) = 5$.

📌 There is exactly one real root in $(-1, 0)$, since $f(-1) < 0$, $f(0) > 0$.

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