I can think of various things but that's not the point! You've been supplied with something that does not work and in law - whether it is a 'puter or an egg timer - you are entitled to a replacement or your money back.
3 reasons why i would rather fix it. 1) I don't enjoy the "It's not our fault" argument 2) I need it working for next week and don't trust them to get it back in time. 3) I'm hoping it's a simple problem, in which case i don't need to send it back.
It sounds to me like a hardware problem - maybe just a loose connection. Not easy to identify or fix in a laptop. Plus, opening the case will invalidate the warranty ... are you getting my drift? Send it back!
With a Laptop you plug in battery - then the AC adapter & turn on the mains where it is plugged in - lift up the lid - press the on button for about 3-5 second and it will boot up. If it does not there is a fault. With the paperwork you received with it there should be a trouble shooting guide with steps to take. If not or they do not work the only other option it to contact the vendor.