Find the number of four digit numbers which can be formed using the digits {0,1,1,2,3,4}

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

Discover how to calculate the number of four-digit numbers that can be formed from the digits {0,1,1,2,3,4}, considering constraints like not starting with 0 and handling repetition of digits. This problem is suitable for students in Grades 9-12 and involves permutation and combinatorics concepts.

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