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The number of such rectangles is always infinite.

Select any length, B such that 0 < B ≤ P/2 units of length and let L = P/2 - B.

Then a LxB rectangle has perimeter 2*(L+B) = 2*(P/2 - B + B) = 2*(P/2) = P.

B was chosen arbitrarily from an infinite number of possible values and so there are infinitely many possible solutions.

See discussion section for further thoughts.

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Q: What is a rule for finding the number of rectangles of different shape that have a given perimeter?
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