Transferring Microsoft Office 2010 to a new computer

I recently bought a new computer, and would like to transfer Microsoft Office (Home and Student) 2010 to my new computer. I Googled it, but couldn't find anything useful (except for apologies that this could not be done without an activation code).
My old computer has Windows 7 Starter, my new computer has Windows 8.1 (if that matters).
I still have the activation code product key, and I also have Office Home and Student installed on two other computers, so I can't use the '3-for-1' deal.

How can I transfer Office to my new computer?

Answer

I called Microsoft (actually, just requested a callback from the AnswerDesk here), and had the issue resolved in 35 minutes.

We just installed Office on my new computer, and put in the product key, and it now works!