Probability of flopping gutshot

[UtzChips]

I am attempting to determine the odds of flopping a gutshot straight draw to 9 [img]/images/graemlins/heart.gif[/img]5 [img]/images/graemlins/heart.gif[/img] without any 4 flushes or pairing my 9 or 5.

This is what I came up with:

The possible Flops are: 87x,86x,76x, where x is not a 9 or 5 or the 3rd card that completes the straight and neither of the inside straight cards are a heart when x is a heart.

1st problem: the inside straight cards are not a heart, so x can be a heart:

3 8s * 3 7s = 9

There are then 48 unseen cards, 6 are 9s & 5s, and 4 are the card that makes a straight on the Flop, so there are 38 cards possible for x.
9*38 = 342 * 3 (because you can get it with 87,86 or 76) =
1026 flops.

2nd problem:
one of the two straight cards is a heart, so x cannot be a heart.
4 8s * 3 7s = 12
there are 10 more suited cards, so 38-10 = 28
12*28*3 = 1008 flops, where one of the straight cards is a heart, but x is not, nor is it a 9 or 5, or the card that completes the straight.

1026 + 1008 = 2034 possible flops
[19,600 - 2034]/2034 = 8.6362:1 odds of flopping a gutshot straight draw without a flush draw, or pairing my 9s or 5s.

I'm 99.9% sure this is right. Can someone verify please.

Thanks.

[Tom1975]

Your method is flawed because it double counts many boards. Problem 1 and Problem 2 are not mutually exclusive so you can't just add their results.