Hi Maddy,
Welcome to the Adobe Forums!
If you downloaded the product via the Creative Cloud, you won't be able to activate the product with a serial number.
To install Adobe Photoshop Lightroom 5 without the Creative Cloud, please follow the link below:
http://www.adobe.com/cfusion/tdrc/index.cfm?product=photoshop_lightroom
If still you have done these steps already and the serial number can't activate the product, please let me know what is the exact error message.
Thank you.
Arnaud.