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📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 Solution: We are to compute the probability that exactly 2 out of 5 students receive their originally assigned experiment (a fixed set of permutations), with the remaining 3 elements forming a derangement with no fixed points among the other students.

📌 Step 1: Total number of permutations of 5 experiments is $5! = 120$.

📌 Step 2: Count the number of permutations where exactly 2 students get their correct experiment.

📌 For the remaining 3 students, we require that none receive their correct experiment — i.e., a derangement of 3 elements. The number of derangements $D_3 = 2$.

📌 Question: A historian of science is examining three independently verified primary source manuscripts from the 17th century. Each manuscript has a 60% chance of containing a previously unknown scientific observation. What is the probability that at least one manuscript contains such an observation?

📌 Question: Jackson rolls four fair 6-sided dice. What is the probability that exactly two of the dice show a 4, and the other two show different numbers, neither equal to 4 nor to each other?

📌 Solution: We compute the probability of rolling exactly two 4s, one distinct number (not 4), and one distinct number (not 4 or the first).

📌 Step 3: Assign the two values to the two remaining dice: $2! = 2$ ways.

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