1. **State the problem:** Find the surface area of a rectangular prism with length $15$ cm, height $7$ cm, and width $5$ cm. 2. **Formula:** The surface area $S$ of a rectangular prism is given by: $$S = 2(lw + lh + wh)$$ where $l$ is length, $w$ is width, and $h$ is height. 3. **Substitute the values:** $$S = 2(15 \times 5 + 15 \times 7 + 5 \times 7)$$ 4. **Calculate each product:** $$15 \times 5 = 75$$ $$15 \times 7 = 105$$ $$5 \times 7 = 35$$ 5. **Sum the products:** $$75 + 105 + 35 = 215$$ 6. **Multiply by 2:** $$S = 2 \times 215 = 430$$ 7. **Final answer:** The surface area of the rectangular prism is $430$ square centimeters.