jmc

Consider two cases:

$\bullet$ Case 1: the person wears a red shirt and a green hat. There are $5$ options for shirts, $6$ options for pants, and $8$ options for hats, for a total of $5\times 6\times 8=240$ outfits.

$\bullet$ Case 2: the person wears a green shirt and a red hat. There are $5$ options for shirts, $6$ options for pants, and $8$ options for hats, for a total of $5\times 6\times 8=240$ outfits.

Summing up the two cases, we have a total possibility of $240+240=\boxed{480}$ outfits.

Notice that we can tackle this without considering cases. The person has $10$ choices for shirts. After a shirt is chosen, the person has $6$ choices for pants, but only $8$ choices of hat, since the hat color cannot match the shirt color, and no matter what shirt color is chosen, there are $8$ hats of the other color to choose from. So, there are $10\times 6\times 8=\boxed{480}$ outfits.