Step-by-step Solution:

\[ \text{Digits: } \{0,1,2,3,4,5\} \] \[ \text{Total sum} = 0+1+2+3+4+5 = 15 \equiv 0 \pmod{3} \] \[ \text{Sum of chosen 5 digits must be divisible by 3} \] \[ \text{So the omitted digit must also be } \equiv 0 \pmod{3} \] \[ \text{Digits divisible by 3: } 0,3 \] \[ \text{Case 1: Omit } 0 \Rightarrow \{1,2,3,4,5\} \] \[ \text{All permutations valid (no leading zero issue)} \] \[ 5! = 120 \] \[ \text{Case 2: Omit } 3 \Rightarrow \{0,1,2,4,5\} \] \[ \text{Total permutations} = 5! = 120 \] \[ \text{Numbers starting with 0} = 4! = 24 \] \[ \text{Valid numbers} = 120 - 24 = 96 \] \[ \text{Total} = 120 + 96 = 216 \] \[ \boxed{216} \]