1. **State the problem:** We want to find the probability that exactly 5 out of 25 randomly selected loans are minority loans, given that the bank claims 10 out of 100 loans are minority loans. 2. **Identify the distribution:** This is a hypergeometric probability problem because we are sampling without replacement from a finite population. 3. **Formula:** The hypergeometric probability is given by $$P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$$ where: - $N = 100$ (total loans), - $K = 10$ (minority loans in population), - $n = 25$ (sample size), - $k = 5$ (number of minority loans in sample). 4. **Apply values:** $$P(X = 5) = \frac{\binom{10}{5} \binom{90}{20}}{\binom{100}{25}}$$ 5. **Interpretation:** This formula calculates the probability of selecting exactly 5 minority loans in a sample of 25, assuming the bank's claim is true. **Final answer:** $$\boxed{P(X=5) = \frac{\binom{10}{5} \binom{90}{20}}{\binom{100}{25}}}$$