Question:
Joe is in his hunting blind when he locates $20$ geese, $25$ ducks, $40$ eagles, $10$ cranes, and $5$ flamingos. Joe randomly selects six birds to target, what is the probability that at least one of each species is targeted?
The correct answer is apparently $0.03985$ according to the textbook solutions.
Work so far:
${20 \choose 1}{25 \choose 1}{40 \choose 1}{10 \choose 1}{5 \choose 1}{95 \choose 1} = 95,000,000$
So there are $95,000,000$ total ways to select at least one of each species.
$95,000,000 \left/ {100 \choose 6}\right. = \frac{95,000,000}{1,192,052,400} = 0.0797$
Notes:
I'm obviously doing something wrong calculating the number of ways to select at least one of each species, as ${100 \choose 6}$ should be the total ways to randomly select 6 of the 100 birds. Obviously my result is the correct answer multiplied by 2, but I'm not sure why. Any help/hints are greatly appreciated.
EDIT: Dividing my 2 should not normally be done in the solution to a problem like this, therefore, I obviously didn't do my calculation correctly. I'd like to see how the calculation would look if I had done it correctly.