[ramjam]

I think your answer is very clever but a bit too clever for me! Also, I think there might be a slight problem in that your formula for a Q flopping is not for just one Q flopping but for one or more Qs flopping - and the same for the 7 formula. As you point out, Copernicus's estimate has a similar problem but to much a larger extent.

I think an easy way to calculate this piece of the puzzle accurately is as follows:
-first card on flop must be a Q, 7 or 6 (6 cards out of 46 unknown)
-second card must match one of the other two pocket pairs (4 out of 45)
- third card must match the final pair (2 cards out of 44).

This gives 6.4.2/(46.45.44) = roughly 0.05% or 1,896 to 1 (very close to your answer - not sure why the odds my way are slightly shorter - there must be some other factor at play).

Anyway if you combine this with the 2.44% estimate of 3 pocket pairs seeing the flop it takes you up to massive 77,000+ to 1!