1. **Problem:** Find the probability that the secretary is female and the chairman and treasurer are of different genders. 2. **Given:** 15 males, 10 females, total 25 people. 3. **Positions:** chairman, treasurer, secretary (3 distinct positions). 4. **Step 1:** Total ways to choose 3 people for the positions: $$25 \times 24 \times 23 = 13800$$ 5. **Step 2:** Secretary must be female: choose 1 female for secretary: $$10$$ 6. **Step 3:** Chairman and treasurer must be of different genders: - Chairman male, treasurer female (excluding secretary): $$15 \times (10 - 1) = 15 \times 9 = 135$$ - Chairman female (excluding secretary), treasurer male: $$(10 - 1) \times 15 = 9 \times 15 = 135$$ 7. **Step 4:** Total favorable ways: $$10 \times (135 + 135) = 10 \times 270 = 2700$$ 8. **Step 5:** Probability: $$\frac{2700}{13800} = \frac{\cancel{2700} \times 1}{\cancel{2700} \times 5.1111} = \frac{1}{5.1111}$$ Simplify fraction: $$\frac{2700}{13800} = \frac{2700 \div 300}{13800 \div 300} = \frac{9}{46}$$ **Final answer:** $\boxed{\frac{9}{46}}$ which corresponds to option (E).