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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 An oceanographer rolls four fair 6-sided dice. What is the probability that exactly three of the dice show the same number?

📌 First, choose the number that appears three times. There are 6 possible numbers (1 through 6):

📌 Next, select which 3 of the 4 dice show this number. The number of ways to choose 3 dice out of 4 is:

📌 The remaining die must show a different number. There are 5 possible values for this die (any number except the one chosen for the three dice):

📌 The total number of outcomes when rolling 4 dice is:

📌 A historian of science has a circular table with 10 different historical artifacts. In how many distinct ways can they be arranged around the table if two specific artifacts must not be adjacent?

📌 Now, calculate the number of arrangements where the two specific artifacts (say A and B) are adjacent. Treat A and B as a single block, reducing the problem to arranging 9 items (the block + 8 others) in a circle:

📌 Within the block, A and B can be arranged in 2 ways (AB or BA):

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