L. Scott Johnson is mostly correct except for 2 things:
- You specified it wasn't a straight flush (there are 7 8-card straight flushes per suit he should have removed.)
- You asked about the 7 card flush, which you were dealt before the draw, not the 8 card flush you drew into.
This gives us:
13C7 as the nCr we care about, giving us 1716 possible flushes of 7 cards.
Then we remove the 8 7-card straight flushes in each suit to give us 1708 possible flushes of 7 cards per suit.
We multiply that result by 4, since we want a flush of 7 cards, not specifically caring about suit giving us 6832 possible non-straight 7 card flushes.
Now we need 52C7, the number of combinations of 7 cards in 52 cards, which is 133,784,560.
This gives us 6832 out of 133,784,560 different 7 card hands that gives us odds of 1194444:61 against, or a 0.005107% chance of such a 7 card flush on the deal (or 0.001277% if you want a specific suit, or odds of 4777959:61 against)
If we do include the 7-card straight flushes, we have 6864 combinations in a deck of 7-card flushes, which gives us odds of 643162:33 against or 0.005131% of possible combinations.