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No matter how many times I do number 20, I end up with 615. Am I wrong, or is the solution sheet wrong?

I think you're right and the given answer of 596 is wrong. It's 380 m-w, 190 m-m, 45 w-w handshakes, so 615 in total.

Yeah. It's not even the only wrong answer on the page, which is troubling.