I don't know about how many there actually are, but you can count the ones that they have listed. And they have repeated a number of countries, but there are enough countries given to consider that remembering a country that way isn't very helpful, at least not to me. I have aced several countries on the first go round that took me multiple attempts to nail on the second (or third) offering, but the reverse is true as well.
What actually helps hugely are the direction and distance hints they give, coupled with a passable general knowledge of world geography. I got a globe and a world wall map for my tenth birthday, so I might be a little more savvy than the average bear in that regard because I've always been interested.
What kicks my ass is the cluster of smaller West Africa countries, and the island countries. My biggest problem sometimes is I can't remember a country's name when I'm trying to pick it from the list. CRS disease. I have had a problem with not recognizing a newer name of some countries. So it's definitely a learning experience. That's why I keep playing.
If I have only the outline of a country but it is shown in the context of the continent it's on and with the surrounding country outlines, I would rarely miss. As it is, I think it's a reasonable test allowing just enough luck to keep it worthwhile and interesting.
Just curious; are you playing, or just kibitzing?