gaming pc on a tight budget

my fourteen year old wants a gaming pc, he has been saving for a while and has about 400 though we can add to it if nescessary about 100. we dont need a moniter and a cheap key board is fine. we just need a good mouse with it
he wants a desktop not a laptop
main game he plays is league of legends recommended specs are
3 GHz processor
2 GB of RAM (4 GB of RAM for Windows Vista and higher)
12 GB available hard disk space
Nvidia GeForce 8800/AMD Radeon HD 5670 or equivalent video card (Dedicated GPU with 512MB or higher Video Memory(VRAM))
Support for DirectX v9.0c or better.
Windows XP SP3, Windows Vista, Windows 7, Windows 8.1, or Windows 10 with the latest service pack installed
The latest update to .NET Framework from Microsoft
The latest version of Adobe AIR is required to connect to PVP.net. Adobe AIR is included with all League of Legends setup files, but can also be downloaded from Adobe's website.

is this totally unobtainable on his budget or is it doable

thank you in advance
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