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📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 To form a unique pair consisting of one star type and one galaxy type, we can use the basic principle of counting. For each star type, there are 5 choices of galaxy types, and vice versa. Therefore, the total number of unique pairs is given by multiplying the number of star types by the number of galaxy types:

📌 To determine how many distinct teams of 3 species can be formed from 8 species, we use combinations since the order of species in a team does not matter. The number of combinations of 8 species taken 3 at a time is calculated using the binomial coefficient:

📌 Thus, the number of different teams is:

📌 We are to seat 5 students and 3 professors around a circular table such that no two professors are adjacent. First, fix one students position to eliminate rotational symmetry (since circular arrangements are rotationally equivalent). Then arrange the remaining 4 students linearly relative to the fixed student.

📌 After fixing one student, the other 4 students can be arranged in $4!$ ways. These 5 students (one fixed + 4 arranged) create 5 possible gaps (one between each pair of adjacent students) where professors can sit. We must choose 3 of these 5 gaps to place one professor each, ensuring no two are adjacent.

📌 Arrange remaining 4 students: $4! = 24$

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