When you say you have a product code for Office 2010, was that pre installed on another computer / activated from the pre installed software or is that from a retail packaged version with the install DVD?
If you have the retail version with an install DVD, just uninstall 2013 and install 2010 using the code from the retail package.
If your code was from the previous laptop and the software came with the laptop or the code was activated from preinstalled software it is likely a a version of office tied to that one computer and cannot be transferred.
$139 for Office 2013 student is about the best I see on the web.
You can get a Keycode for Office 2010 (no media) for $99 then you can download the install from Microsoft.
Finally if you want to try it, download the office 2010 Media from Microsoft, uninstall the 2013 from the current laptop and see if your office 2010 will activate with your keycode.
To Download the 2010 install goto: office . com / downloadoffice (Remove the spaces)