1. **Stating the problem:** We have a symphony orchestra with 30 Haydn symphonies, 15 modern works, and 9 Beethoven symphonies. The program consists of one Haydn symphony, followed by one modern work, and then one Beethoven symphony. We want to find how many different programs can be played. 2. **Formula used:** The total number of different programs is the product of the number of choices for each part of the program because each choice is independent. So, the formula is: $$\text{Total programs} = (\text{number of Haydn symphonies}) \times (\text{number of modern works}) \times (\text{number of Beethoven symphonies})$$ 3. **Applying the numbers:** $$\text{Total programs} = 30 \times 15 \times 9$$ 4. **Calculating:** $$30 \times 15 = 450$$ $$450 \times 9 = 4050$$ 5. **Answer:** The orchestra can play $$\boxed{4050}$$ different programs.