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Home » Solution: Factor out the greatest common factor, $6x$: - IQnection

Solution: Factor out the greatest common factor, $6x$: - IQnection

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 There are 2 such outcomes (5, 6) out of 6, so:

📌 We want exactly 2 out of 4 dice to show a number > 4. This is a binomial probability:

📌 Question: At a quantum physics seminar, 6 researchers agree to sit in a circle to discuss wave-particle duality. Two of them, Dr. Lee and Dr. Kim, insist on sitting directly opposite each other. In how many distinct arrangements can the group be seated, considering rotations as identical?

📌 We are arranging 6 people in a circle such that Dr. Lee and Dr. Kim sit directly opposite each other.

📌 Fix Dr. Lee at position 1. Then Dr. Kim must be at position 4 (directly opposite). This fixes two positions.

📌 The remaining $4$ researchers can be arranged in the remaining $4$ seats: $4! = 24$ ways.

📌 Since we are counting circular arrangements and we accounted for rotational symmetry by fixing Dr. Lees position (a standard technique), and we required a fixed opposition, this method is valid and avoids overcounting.

📌 Number of ways to place Dr. Lee: 6 choices.

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