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Home » One of these integers is divisible by 5. - IQnection

One of these integers is divisible by 5. - IQnection

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: A pharmacologist is testing combinations of 3 neuroprotective drugs, 2 anti-inflammatory agents, and 1 cognitive enhancer from a pool of 6 neuroprotective, 5 anti-inflammatory, and 4 cognitive drugs. How many distinct treatment combinations can be formed if the order of selection does not matter?

📌 Question: A geographer is analyzing satellite images of a coastal region divided into 15 zones, where sea level rise has affected exactly 6 of them. If the regions are indistinct except by location, how many ways can the affected zones be selected such that no two adjacent zones are both affected, assuming the zones are arranged in a straight line?

📌 Solution: We model this as placing 6 affected zones (A) among 15 positions, such that no two As are adjacent. To ensure non-adjacency, place 6 As with at least one unoccupied (U) zone between them. First, place 6 As with one U between each pair: this requires $6 + 5 = 11$ positions, leaving $15 - 11 = 4$ Us to be freely distributed in the $7$ gaps (before the first A, between As, and after the last A). This is a stars and bars problem:

📌 Question: Jackson rolls four fair 6-sided dice, each numbered from 1 to 6. What is the probability that exactly two of the dice show a prime number?

📌 We want exactly 2 dice to show a prime. This is a binomial probability:

📌 Now convert all to denominator $156250$ for consistency:

📌 Now convert all to denominator $15625$:

📌 Total favorable: $4096 + 4913 + 4913 + 27744 = 41666$

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