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The total number of Pokemon in the game is 493. The amount you can catch with just Platinum is around 440. The other 50 or so Pokemon you'll need to trade, get from events or import via Pal Park.
Cool. That's way more than I thought. I thought it would just be about 200 or so. Also I read somewhere that you could catch the legendary birds in Platinum, is that true or are those part of the Pokémon I would have to trade in?
If by legendary birds, you mean Articuno, Zapdos, and Moltres, then yes, you can catch them without trading.
Don't open if you don't want to be spoiled:
Spoiler:
After recieving the National Dex, you have to go to Eterna City and talk to Prof. Oak. This will trigger the event that causes them to start roaming around Sinnoh.
Ironically from your avatar, Zapdos is the last one I need to catch on Platinum. He is at 1 hp using false swipe and paralyzed and I still can't get him... but I will.
I think this question thread would have been better posted, but instead of merging, I'll close the thread. If you feel unhappy with the responses in this thread for how many Pokemon can be caught in Platinum, please post your question in the Official Generation IV Question Thread. ^^