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How to drill big holes. Grey areas = holes in the wood.

Start by marking where the "cutting" should be done. (1)
Drill holes, close to each others, edge-to-edge with the inside of the marking. (2)
Hit the middle so it falls out. Now it looks like in (3).
Use a file or buffing rasp to even the edges, so that the hole becomes round and nice.
This doesn´t take many minutes, and as you can see -- you don´t need a saw for this.