how do I find the serial code for the Sims 3 that I downloaded off Origin?

When you purchase The Sims 3 or any of its Expansions or Stuff Packs through Origin, you no longer need to manually register your product through The Sims 3’s website. Just make sure that your Origin account details match the details of your Sims 3 account, and Origin will register your products for you.

If you don't already have a Sims 3 or Origin account, you can make one by going to http://www.thesims3.com 

Update: If you already have an Origin account and bought The Sims 3 or any of its Expansions or Stuff Packs through Origin, you do not need to create an account on the Sims 3 website. Just log in with your Origin account and you should be able to see your products on your "My Page" website.

Read more about that here:

http://answers.ea.com/t5/The-Sims-3/Information-How-to-Register-The-Sims-3/m-p/387162