|MadSci Network: Astronomy|
Like a lot of questions in astronomy, yours has many answers! It depends on how you are looking at the stars.
Modern astronomers have well defined boundaries for each constellation. I just did a quick search on a database, and found that there are about 250 stars visible to the naked eye inside the borders of what we call "Orion". However, from where I am, I can rarely see more than about 20! That's because there are lots of lights where I am, and the sky gets washed out.
But that's only with your naked eye. Time exposures using cameras also can show much fainter stars. If you look at any basic astronomy textbook I'm sure you'll find some long exposures of Orion. Even with a small camera you can see thousands of stars in a one hour exposure!
Stars can even be so faint you need a telescope to see them, and the bigger the telescope, the fainter you can go. Using the biggest telescopes in the world, I am sure you can see many millions of stars inside Orion's borders.
Not only that, but there are many stars that are blocked by gas and dust in our Galaxy, If we could sweep that away, you would see billions of stars! And finally, there are lots of other galaxies, extremely faint, in Orion, each containing many billions of stars. There are perhaps trillion or quadrillions of stars in Orion! It just depends on how faint you want to go.
Try the links in the MadSci Library for more information on Astronomy.