We are tasked with finding the number of four-digit numbers that can be formed using the digits $\{ 0, 1, 1, 2, 3, 4 \}$.

Step-by-step analysis:

1. Digits and Constraints: We have the digits 0, 1, 1, 2, 3, 4 available, but we cannot start a four-digit number with 0.

2. Case 1: No restrictions on the first digit: If there were no restrictions (i.e., if we were allowed to use any of the 6 digits for any position), we could calculate the number of four-digit combinations as:

   • We have 6 digits: $\{ 0, 1, 1, 2, 3, 4 \}$.
   • However, we must account for the repetition of the digit '1'.

   If we treat the digits as unique (ignoring the repetition), the number of ways to arrange four digits from 6 would be $P(6,4) = 6 \times 5 \times 4 \times 3$.

3. Case 2: Exclude starting with zero: Since four-digit numbers can't start with zero, we must adjust our calculation. We consider two sub-cases based on the digit in the first position.

4. Sub-cases based on starting digit:

   • If the first digit is 1, 2, 3, or 4: There are 4 options for the first digit. For the remaining three positions, we can use the remaining 5 digits (including 0) but must account for the repetition of '1'.

     We can solve this by considering whether the number uses one or two of the '1's:

     • Use all digits inclusively: Steps: ==--> Instead clarify