Can I run #10 and #12 cables through the same hole in framing lumber?
In the US, as far as the National Electrical Code is concerned, you can run three cables through a hole bored in a joist or rafter as long as you can run the cables without damaging them.  

If the bored hole was longer than 24", you would have to observe the rules for bundling and do derating calculations - that is, figure out the diminished current carrying capacity of the cables.  This would be very unusual in house construction, there are not many walls, beams, etc. that are 24" deep.  

There may be local codes that apply in addition to or instead of the NEC - can't say what applies for your area.  

Some people have their own rules about how many cables can go through a single bored hole.  Sometimes these rules of thumb are pretty popular, and can be confused with actual code requirements, even by inspectors.  

The safest thing to do in these situations is contact your local code enforcement department and ask.  Of course, it's often a lot easier to drill another hole.