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The number of ways to choose 2 signatures from 5 is: - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Let $ x_1 + x_2 + x_3 + x_4 = 6 $, where $ x_i \geq 1 $.

📌 Substitute $ y_i = x_i - 1 $, so $ y_i \geq 0 $ and:

📌 y_1 + y_2 + y_3 + y_4 = 6 - 4 = 2

📌 Number of non-negative integer solutions to this equation is:

📌 \binom{2 + 4 - 1}{4 - 1} = \binom{5}{3} = 10

📌 Hence, the number of valid installation configurations is $ \boxed{10} $.

📌 Question: A deep-sea researcher collects 9 extremophile microorganisms near a hydrothermal vent, including 4 thermophiles, 3 psychrophiles, and 2 barophiles. If the researcher sequences DNA samples in a lab dish by selecting one organism per day for 9 consecutive days, with organisms of the same type indistinguishable, how many distinct sequencing orders are possible?

📌 Solution: We are arranging 9 organisms where there are repeated types: 4 thermophiles (T), 3 psychrophiles (P), and 2 barophiles (B). The number of distinct permutations of a multiset is given by:

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