How many countries run through the river Rhine?

The Rhine starts in Switzerland, flows to the German border, forms the border between Germany and France, flows into Germany, then to the Netherlands, and finally into the North Sea. So, counting France, that's four.

It also borders on Liechtenstein and Austria, so that makes 6.

But if you are being pedantic about "run through" and excluding countries it merely borders, then the answer is 3.