Thank you, that was indeed the solution. And thank you for being polite to an old lady who is somewhat inept at all things computer.
However, Live Chat still would not give me a product code, even after I photographed and uploaded both the box and the receipt. They suggested I purchase the game online from Origin, in which case they would be able to address any issues, and when I protested that I had already paid $60 for the game and wasn't going to re-purchase it, I got essentially a polite "can't help you then" shrug from the Chat guy.
So I will be returning the game to the store, and EA has just lost $60 and a customer. If they can't be bothered to have Quality Control make sure there are product codes in every box, they don't deserve my business, tiny drop in the bucket of their annual bottom line though it may be.
And, needless to say, I won't be using the 15% discount code on my next purchase from Origin that he emailed me, tyvm.
Thanks again for all your help.