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Choice C is correct. The perimeter of a quadrilateral is the sum of the lengths of its four sides. It's given that the rectangle has a length of $13$ and a width of $6$. It follows that the rectangle has two sides with length $13$ and two sides with length $6$. Therefore, the perimeter of the rectangle is $13+13+6+6$, or $38$.

Choice A is incorrect. This is the sum of the lengths of the two sides with length $6$, not the sum of the lengths of all four sides of the rectangle.

Choice B is incorrect. This is the sum of the lengths of the two sides with length $13$, not the sum of the lengths of all four sides of the rectangle.

Choice D is incorrect. This is the perimeter of a rectangle that has four sides with length $13$, not two sides with length $13$ and two sides with length $6$.