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📅 April 18, 2026 👤 scraface

Apr 18, 2026

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📌 We need to determine the number of ways to select 3 states out of 5 to remain unchanged, and then distribute the 2 particles among these states such that the particles occupy exactly 1 of the 3 unchanged states (since they must be identical and indistinguishable).

📌 First, choose 3 states from the 5 to stay unchanged:

📌 For the 2 identical particles, they must be placed in one of the 3 selected unchanged states. Since particles are identical, there is exactly 1 way to assign both particles to the same state (and only one of the 3 states is chosen).

📌 Therefore, the total number of configurations is:

📌 Hence, the number of valid configurations is:

📌 A science policy analyst evaluates funding allocations across 6 scientific disciplines. Each of 4 proposed grants is independently assigned to one discipline. What is the probability that exactly 2 disciplines receive 2 grants each, and the remaining 2 disciplines receive none?

📌 First, compute the total number of ways to assign 4 grants to 6 disciplines:

📌 Now, count the number of favorable assignments where exactly 2 disciplines receive 2 grants each.

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