How to create a DFA for min. one 0 and max. three 1’s

Question

For the following language I need to create a DFA, but automatons I create are always nondeterministic finite ones (I don’t know if it’s correct).

Σ = {0, 1}
L = {ω ϵ Σ* | (|ω|₀ ≥ 1) ∧ (|ω|₁ ≤ 3)}

Answer 1

The solution was to convert the NFA to a DFA via subset-construction as suggested by harold.