8 dice are rolled, what is the probability that you roll 4 identical even numbers and 4 identical odd numbers.
First Odd/Even Roll 6C1 = 6
Next 4 Rolls All Odd/Even 8C4 = 70
Opposite Odd/Even Roll 3C1 = 3
Last 4 Rolls Opposite Odd/Even 4C4 = 1
All Possible Rolls 6^8 = 1679616
$P = (6*70*3*1)/1679616 = 0.00075$
Have I done this problem correctly?
To add to JMoravitz's comments, a way of thinking of this is that there are $3$ choices for the even face and $3$ choices for the odd face.
Then, out of the $8!$ ways to order the $8$ dice, since the even and odd faces are indistinguishable, we have $$ \frac{8!}{4!4!}$$ different orderings. Hence there are $$ 3\cdot3\cdot\frac{8!}{4!4!} = 630 $$ ways for this to happen, and so a probability of $$ \frac{630}{6^8} = \frac{35}{93312} \approx 0.00037508573. $$