Omaha - easy probability question

[Flip-Flop]

Omaha, 3 players see a flop, Hero + 2 vilians.
The flop is suited, lets say spades.
Hero doesn`t have any spades.
What`s the probability that any of the vilians have a flush and what is the easiest/fastest way to calculate this?

I know how to get there but I think I`m over-calculating it.
There has to be a simpler way.
Anyone?

[Buzz]

Flip Flop – Roughly three times out of eight, at least one of your two opponents will have the flush.

If I don’t have to use words to explain what I did, here is the rigorous calculation:

8 spades 45 C(10,8)*C(35,0)
7 spades 4200 C(10,7)*C(35,1)
6 spades 124950 C(10,6)*C(35,2)
5 spades 1649340 C(10,5)*C(35,3)
4 spades 10995600 C(10,4)*C(35,4)
3 spades 38955840 C(10,3)*C(35,5)
2 spades 73042200 C(10,2)*C(35,6)
1 spade 67245200 C(10,1)*C(35,7)
0 spades 23535820 C(10,0)*C(35,8)
total is 215553195
total should be C(45,8)=215553195 checks

8 spades 45
7 spades 4200
6 spades 124959
5 spades 1649340
4 spades 10995600
3 spades 38955840
2 spades 31303800 (from 73042200*30/70)
1 spade 0
0 spades 0
total is 83033784

83033784/215553195 = 0.38521249 = probability

I don’t want to explain it.

I can think of a couple of ways to approximate it well enough for our purposes.

Buzz

[bigpooch]

It's easier to work with the numbers for 0, 1 or 2 spades;
then, just subtract the probability from one.

[Flip-Flop]

Thanks