Introduction:
In this question, we need to find the number of 4-digit numbers that can be formed using the digits 1, 1, 2, 2, 3, and 4. We will solve this problem step by step, considering all the possible combinations and permutations of the given digits.

Step 1: Counting the total number of possibilities:
To find the total number of possibilities, we need to consider the 4-digit numbers that can be formed using the given digits. Since we have 6 different digits, there are a total of 6 choices for each digit. Therefore, the total number of possibilities can be calculated as:

Total number of possibilities = 6 × 6 × 6 × 6 = 6^4 = 1296

Step 2: Counting the number of repeating digits:
In the given digits, we have two 1s and two 2s. We need to consider the cases where both 1s or both 2s are used in the 4-digit number.

Case 1: Both 1s are used:
To form a 4-digit number with both 1s, we need to choose one position for the remaining two digits (2, 3, or 4). There are a total of 3 choices for this position. The remaining two digits can be arranged in 2! = 2 ways. Therefore, the total number of possibilities for this case is:

Number of possibilities = 3 × 2! = 3 × 2 = 6

Case 2: Both 2s are used:
Similar to Case 1, we need to choose one position for the remaining two digits (1, 3, or 4). There are a total of 3 choices for this position. The remaining two digits can be arranged in 2! = 2 ways. Therefore, the total number of possibilities for this case is:

Number of possibilities = 3 × 2! = 3 × 2 = 6

Step 3: Counting the number of unique digits:
In the remaining digits (3 and 4), we have two unique digits. We need to consider the cases where only one of these digits is used in the 4-digit number.

Case 1: Only 3 is used:
To form a 4-digit number with only 3, we need to choose three positions for the remaining three digits (1, 2, or 4). There are a total of 3 choices for each position. Therefore, the total number of possibilities for this case is:

Number of possibilities = 3 × 3 × 3 = 27

Case 2: Only 4 is used:
Similar to Case 1, we need to choose three positions for the remaining three digits (1, 2, or 3). There are a total of 3 choices for each position. Therefore, the total number of possibilities for this case is:

Number of possibilities = 3 × 3 × 3 = 27

Step 4: Calculating the final result:
To find the total number of 4-digit numbers formed using the given digits, we need to subtract the number of repeating digits from the total number of possibilities and add the number of unique digits:

Total number of 4