This is third solution. 1. You might try to repair your client. (Run as Administrator) 2. If that doesn't work, make sure you have .NET Framework 3.5 or 4.
Make sure your computer is up to date, restart your computer and after
that remember to run League of Legends as Administrator.
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One of the best solutions is:
1. Right click your LoL icon
2. Click on properties
3. Click on compabilitity
4. Choose your OS so it actually is yours
5. Click on run as admin