#### 17, 18Question: A palynologist is analyzing the distribution of pollen samples from 5 different plant species labeled A through E. If each sample must be analyzed exactly once and the order of analysis matters, how many distinct sequences are possible if species A must be analyzed before species B? - Coaching Toolbox
Title: How Many Valid Sequences Exist When Analyzing Pollen Samples from Species A to E with A Before B?
Title: How Many Valid Sequences Exist When Analyzing Pollen Samples from Species A to E with A Before B?
When a palynologist analyzes pollen samples from five distinct plant species labeled A through E, the task involves arranging all five samples in a particular order—since the order of analysis matters. Each species’ sample is analyzed exactly once, forming a unique permutation of the five species.
However, there’s a key constraint: species A must be analyzed before species B. This condition changes the total number of valid sequences compared to the unrestricted permutations.
Understanding the Context
Total Permutations Without Restrictions
For five distinct items, the total number of possible sequences (permutations) is:
$$
5! = 120
$$
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Key Insights
Imposing the Condition: A Before B
Among all 120 permutations, only half satisfy the condition that A appears before B, because in any random arrangement of A and B, A is equally likely to come before or after B.
Thus, the number of valid sequences where A is analyzed before B is:
$$
rac{5!}{2} = rac{120}{2} = 60
$$
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Why This Works
Think of all 120 sequences. For every sequence where A comes before B, there’s a corresponding sequence with A and B swapped—where B comes first. These pairs are indistinguishable under the condition. Since each pair of positions for A and B is equally likely, exactly half the permutations satisfy A before B.
This simple symmetry reduces the total number of valid sequences by half.
Conclusion
When analyzing 5 unique pollen samples labeled A through E, where each is analyzed once and order matters, and species A must be analyzed before species B, there are exactly 60 distinct valid sequences.
This reflects the power of combinatorial symmetry—reducing the total permutations by factor of 2 when a relative ordering constraint is imposed.
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