The Shocking Truth: They’re Bringing the Hobbits to Isengard—You Won’t Believe What Happens Next!

The Shocking Truth: They’re Bringing the Hobbits to Isengard—You Won’t Believe What Happens Next!

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Question: A digital content strategist creates a community engagement challenge where participants draw cards from a deck of $3n$ uniquely labeled cards numbering from 1 to $3n$. If $n = 4$, and one participant draws 3 cards at random without replacement, what is the probability that the sum of the card numbers is divisible by 3?Solution: Let $n = 4$, so the deck has $3n = 12$ cards labeled from 1 to 12. We want the probability that the sum of 3 randomly drawn distinct cards is divisible by 3.We analyze residues modulo 3. The numbers from 1 to 12 modulo 3 fall into three residue classes:Total favorable: $4 + 4 + 4 + 64 = 76$Question: A pharmacologist is testing combinations of 4 experimental compounds from a pool of 8, where 3 are neuroprotective and 5 are anti-inflammatory. If the selection includes at least one compound from each category, how many such combinations are possible?We subtract the cases that violate the at least one from each category condition:Also exclude the case where no neuroprotective compound is chosen (already included), and no anti-inflammatory — also impossible.So only invalid case is selecting all 4 from anti-inflammatory: 5 ways.Thus, valid combinations: $70 - 5 = 65$Alternatively, count valid distributions: