1. The problem is to find the surface area of a rectangular prism with length $2$ inches, width $1$ inch, and height $1$ inch. 2. The formula for the surface area $S$ of a rectangular prism is: $$S = 2(lw + lh + wh)$$ where $l$ is length, $w$ is width, and $h$ is height. 3. Substitute the given values into the formula: $$S = 2(2 \times 1 + 2 \times 1 + 1 \times 1)$$ 4. Calculate each product inside the parentheses: $$S = 2(2 + 2 + 1)$$ 5. Add the values inside the parentheses: $$S = 2(5)$$ 6. Multiply by 2 to find the total surface area: $$S = 10$$ 7. Therefore, the surface area of the rectangular prism is $10$ square inches.