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You Won’t Believe How to Teletransport Instantly—Science Has Gone Significant!
2Question: A digital health solutions architect is designing a system that logs patient check-ins in real-time across multiple rural clinics. If the system must assign each patient ID uniquely using a 6-digit code where each digit is from 1 to 9 (no zeros), and no digit is repeated in any code, how many distinct check-in codes can be generated?
Solution: To determine the number of distinct 6-digit codes using digits from 1 to 9 with no repetition, we calculate the number of permutations of 9 digits taken 6 at a time:
P(9, 6) = \frac{9!}{(9 - 6)!} = \frac{9!}{3!}
9! = 362880,\quad 3! = 6
\frac{362880}{6} = 60480
Thus, the total number of unique check-in codes is $\boxed{60480}$.
Question: An industrial hygienist in Los Angeles is analyzing exposure levels in 6 different work zones, each with 3 safety zones. If 2 zones are randomly selected for an immediate inspection, what is the probability that both are in the same work area (i.e., same high-risk level), assuming each work area contains exactly 2 zones?
Solution: There are 6 work areas, each with 2 zones, making a total of $6 \times 2 = 12$ zones. Since 2 zones are selected at random, the total number of ways to choose 2 zones is:
\binom{12}{2} = \frac{12 \times 11}{2} = 66
