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Do I have to complete different Mythic+ dungeons to get multiple great vault choices?
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Does completing the same dungeon at different keys count towards the GV requirement?
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What about completing the same dungeon at the same key multiple times?
No. If you run +7 tirna scithe 4 times, your vault will have 2 rewards at 213 next week.
Yes.
This seems like a rephrasing of your first question, in which case see the first answer.
To 3. - Doing the same dungeon for all of them doesnt restrict the loot table to just that dungeon.