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📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Question: A science administrator is assigning 7 research grants to 3 different programs. Each grant is distinct, and each program must receive at least one grant. How many ways can the administrator distribute the grants?

📌 This is a classic problem of distributing distinct objects (grants) to distinct groups (programs) with no empty group allowed.

📌 We use the principle of inclusion-exclusion or equivalently, Stirling numbers of the second kind combined with permutations.

📌 First, partition the 7 distinct grants into 3 non-empty subsets. The number of such partitions is given by the Stirling number of the second kind $ S(7, 3) $. From known values:

📌 Then, assign each of the 3 subsets to one of the 3 programs. Since the programs are distinct, this can be done in:

📌 Thus, the number of ways to distribute the grants is:

📌 Question: A soil scientist is studying 4 types of bacteria, with 3 samples of each type, for a lab experiment. If she randomly selects 5 samples to analyze, what is the probability that she gets at least one sample from each of 3 different types?

📌 To ensure at least one sample from 3 types, we consider all combinations of types that include 3 types (and possibly a 4th type in the remaining 2 samples). The possible type distributions are:

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