How many satisfying assignments are there in a set of 3-CNF clauses where no clause share the same variable?
As you noted, for every clause there are $2^3$ possible assignments. Exactly one of them does not satisfy the clause: it's the one that has false for all variables that appear in the clause as a positive, and true for the variables that appear as a negative. In your example it is $001$. The rest of the assignments satisfy the clause. Therefore there are $2^3 - 1 = 7$ satisfying assignments for each clause, which means there are $7^k$ possible assignments in total.