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Choice C is correct. The area of a rectangle with length $l$ and width $w$ can be found using the formula $A=lw$. It’s given that the rectangle has a length of $17 \text{cm}$ and a width of $7 \text{cm}$. Therefore, the area of this rectangle is $A=17\left(7\right)$, or $119 {\text{cm}}^2$.

Choice A is incorrect. This is the sum of the length and width of the rectangle, not the area.

Choice B is incorrect. This is the perimeter of the rectangle, not the area.

Choice D is incorrect. This is the sum of the length and width of the rectangle squared, not the area.