1. **Problem Statement:** We have $n$ chicks belonging to $\log n$ hens. Each chick has exactly one mother hen, and a hen can have between 0 and $n$ chicks. We want to find the lower bound on the number of inspections needed to correctly match each chick to its mother hen using a decision-tree approach. 2. **Understanding the Problem:** Each inspection tests if a chick belongs to a hen by placing them together. The goal is to identify the unique mother for each chick. 3. **Key Insight - Decision Tree Model:** The problem is equivalent to identifying a function from $n$ chicks to $\log n$ hens. Each chick's mother is one of $\log n$ hens, so there are $(\log n)^n$ possible assignments. 4. **Number of Possible Outcomes:** The number of ways to assign $n$ chicks to $\log n$ hens is: $$ (\log n)^n = 2^{n \log (\log n)} $$ 5. **Decision Tree Height (Lower Bound):** Each inspection is a binary test (yes/no). To distinguish among all possible assignments, the decision tree must have at least as many leaves as the number of assignments. The height $h$ of the decision tree satisfies: $$ 2^h \geq (\log n)^n = 2^{n \log (\log n)} $$ Taking $\log_2$ on both sides: $$ h \geq n \log (\log n) $$ 6. **Interpretation:** The lower bound on the number of inspections is $\boxed{n \log (\log n)}$. 7. **Check Options:** Among the given options, $n \log \log n$ corresponds to option (f). **Final answer:** (f) $n \log \log n$