One last thing. A 64-bit computer can natively run a 32-bit OS just fine, or it can run a 64-bit OS. However, a 32-bit computer cannot run a 64-bit OS.
One more followup. If a 64-bit computer runs a 32-bit OS, then invariably all the benefits of the 64-bit processor are lost. There are some ways to use add-ons and extensions for some OSes, but in reality, you are stuck with all of the bad, just as if you had a 32-bit processor running a 32-bit OS.