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📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Question: A UX researcher is testing 4 interface designs with 6 participants. If each participant tests exactly one design and each design must be tested by at least one participant, how many ways can the participants be assigned to designs?

📌 Solution: This is a classic surjection problem: distributing 6 distinguishable participants to 4 distinguishable designs such that no design is left untested. The number of such surjective functions is given by:

📌 Question: A science policy expert is organizing a roundtable with 3 government officials and 3 industry leaders. If the seating must alternate government and industry around a circular table, how many distinct seating arrangements are possible?

📌 Solution: In a circular arrangement with alternating roles, we fix one persons position to account for rotational symmetry. Fix a government official at one seat. Then the seats alternate: G, I, G, I, G, I. The remaining 2 government officials can be arranged in $2!$ ways, and the 3 industry leaders in $3!$ ways. So the total number of distinct arrangements is:

📌 Question: A UX researcher is analyzing click patterns on a webpage with 5 buttons labeled A through E. If a user clicks exactly 4 buttons in sequence, with no repetition, how many such sequences include both button A and button B?

📌 Question: A chemist is designing a reaction that requires choosing 3 out of 7 available catalysts, where exactly one catalyst must be a metal-based type (and there are 3 such catalysts available). The rest must be non-metal-based (4 available). How many valid combinations of catalysts can be used?

📌 Question: A science policy expert is forming a panel of 4 experts from a group of 6 scientists and 5 policymakers. What is the probability that the panel includes at least 2 scientists and at least 1 policymaker?

📌 We count favorable cases where there are at least 2 scientists and at least 1 policymaker. The valid compositions are:

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