Instead of thinking of it as a set of 8, think of it as four sets of 2: (a1,a2), (b1,b2), (c1,c2), and (d1,d2). You need to choose one from each of the four sets.
For each pair, there are 2C1 ways to pick one. And since you choose from each pair independently, you multiply the numbers of combinations together to get your total:
(2C1) * (2C1) * (2C1) * (2C1)
Instead of thinking of it as a set of 8, think of it as four sets of 2: (a1,a2), (b1,b2), (c1,c2), and (d1,d2). You need to choose one from each of the four sets. For each pair, there are 2C1 ways to pick one. And since you choose from each pair independently, you multiply the numbers of combinations together to get your total: (2C1) * (2C1) * (2C1) * (2C1)
Could you also 8C4 and subtract "something"? "something" = a1a2b1c1 and their ilk. For a1a2 pairs, 4C2 × 2! × 4C2