Solution:

Using the Pythagorean theorem, we get that $$P'Q' = \sqrt{3^2 - 1^2} = 2\sqrt{2}$$ and $$Q'R' = \sqrt{5^2 - 1^2} = 2\sqrt{6}.$$

This follows from $PQ = 1 + 2 = 3$ and $QR = 2 + 3 = 5.$ The heights of the triangles are also just $1.$

Then, we get that $$[Q'QPP'] = \dfrac{1}{2}(1 + 2)2\sqrt{2}$$$$= 3\sqrt{2}.$$

We also get that $$[R'RQQ'] = \dfrac{1}{2}(2 + 3)2\sqrt{6}$$$$= 5\sqrt{6}.$$

Finally, we have that $$[R'RPP'] =$$$$\dfrac{1}{2}(1 + 3)(2\sqrt{2} + 2\sqrt{6})$$$$= 4\sqrt{2} + 4\sqrt{6}.$$

Now, we can express $[PQR]$ as $$[Q'QPP'] + [R'RQQ']$$$$- [R'RPP'].$$

This evaluates to $$3\sqrt{2} + 5\sqrt{6} - 4\sqrt{2} - 4\sqrt{6}$$ $$= \sqrt{6} - \sqrt{2}.$$

Thus, the correct answer is D .