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5Question: An entomologist studies 6 species of bees and 4 species of butterflies. If she wants to select 3 species for a pollination experiment, with at least 1 bee and 1 butterfly, how many distinct groups can she form?
Question: An archaeologist discovers 7 distinct pottery shards from a Peruvian site and 5 from a Mesoamerican site. If she arranges them in a row on a shelf with all Peruvian shards indistinct and all Mesoamerican shards indistinct, how many distinct arrangements are possible?
Solution: The problem reduces to permutations of a multiset. There are $ 7 + 5 = 12 $ total shards, with 7 identical Peruvian and 5 identical Mesoamerican. The number of distinct arrangements is $
Question: A hydrologist models groundwater flow through 3 layers of soil, each with 2 possible flow channels (A or B). If the system must have exactly 2 channels flowing vertically and 1 horizontally, how many distinct flow configurations are possible?
Question: An entomologist tracks 5 types of pollinators visiting 3 flower species. If each pollinator visits exactly one flower per day for 2 days, how many distinct 2-day visitation sequences are possible?
Question: An archaeologist excavates 4 stone tablets and 3 ceramic vessels. If she displays them in a row with the tablets indistinct among themselves and the vessels indistinct among themselves, how many distinct arrangements are possible?
Solution: The problem is equivalent to arranging 4 identical tablets and 3 identical vessels. The number of distinct arrangements is $
Question: A hydrologist simulates water movement through 6 underground aquifers, each with 2 possible directions (left or right). If the system must have exactly 4 leftward flows and 2 rightward flows, how many distinct flow patterns are possible?
A lunar regolith extraction engineer needs to determine how many two-digit integers can evenly distribute the weight of a mining vehicle that needs to hold on average 13.5 kilograms. These integers must be divisible by 3. How many such integers exist?
To solve the problem, we need to find the two-digit integers that are divisible by 3. A two-digit integer ranges from 10 to 99.