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Home » 2x = 6 \quad \Rightarrow \quad x = 3 - NBX Soluciones

2x = 6 \quad \Rightarrow \quad x = 3 - NBX Soluciones

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A self-healing concrete restoration project in Tokyo uses $4n$ microcapsules, each activated in a sequence determined by a random permutation of the numbers $1$ through $4n$. What is the probability that in a randomly selected sequence, the first $n$ microcapsules all consist of unique chemical types, assuming there are exactly $n$ distinct types, with 4 identical copies of each type?

📌 Total number of distinct permutations of the $4n$ microcapsules, accounting for repetitions, is:

📌 since each type has 4 identical copies.

📌 Now, count the favorable permutations where the first $n$ positions contain one copy of each of the $n$ types.

📌 Choose one copy of each type for the first $n$ positions: since the copies are indistinct within a type, but we must assign one distinct type per position, there are $n!$ ways to assign the $n$ distinct types to the first $n$ spots.

📌 The remaining $3n$ positions must be filled with the remaining $3$ copies of each type. The number of ways to arrange these is:

📌 since each type now has 3 identical copies left.

📌 Thus, the number of favorable permutations is:

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