Leave 10 out! Consider {2,4,8}, {3,9}, {5}, {7}.
Now to have copimes we must chose at least two coming from different sets.
Say: {2,9,7} or {5,7}. How many such sets are there?
On the other hand, do you mean “how many pairs of coprimes are there?”
If so, then count the ordered pairs.
Now in either case, go back and see where you can ‘add’ 10.