To determine how many four-digit numbers we can create from the digits 1, 1, 1, 2, 2, 3, we must consider the following steps:

1. Total Digits Available: The digits available are 1, 1, 1, 2, 2, 3.
2. Selection of Four Digits: We need to select 4 digits from these six. The selection will depend on how many times each digit can be used.

Let's break it down:

• Case 1: All four digits are different.

  • This is not possible since we only have three distinct digits (1, 2, 3).

• Case 2: Three digits are the same, and one is different.

  • This is not possible since the digit "1" is the only digit that appears three times, and we need four digits total.

• Case 3: Two digits are the same, and the other two digits are different.

  • The only combination that satisfies this is choosing two '1's and two other distinct digits.
  • For example, we can select {1, 1, 2, 3}.

• Case 4: Two digits appear twice each.

  • This is also possible. We can select two '1's and two '2's.

Calculation

1. For the combination {1, 1, 2, 3}:

   • The number of ways to arrange these four digits is calculated by: $\text{Number of permutations} = \frac{4!}{2!} = \frac{24}{2} = 12$

2. For the combination {1, 1, 2, 2}:

   • The number of ways to arrange these four digits is calculated by: $\text{Number of permutations} = \frac{4!}{2! \times 2!} = \frac{24}{4} = 6$

Total Number of Four-Digit Numbers

• Add the results from the two cases: $\text{Total} = 12 + 6 = 18$

Thus, 18 four-digit numbers can be created using the digits 1, 1, 1, 2, 2, 3.

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Do you want further details or have any questions? Here are 8 related questions to consider:

1. How would the answer change if the digits were 1, 1, 2, 2, 3, 3?
2. Can you find the number of distinct five-digit numbers possible with the same digits?
3. What if all digits were distinct? How many four-digit numbers could be formed then?
4. How would the number of four-digit numbers change if we had only one '1' and the other digits the same?
5. How would the solution change if repetitions were not allowed?
6. What are permutations and combinations, and how are they different?
7. Can you explain why factorials are used in permutation calculations?
8. How does the presence of repeated digits affect the total number of permutations?

Tip: When counting permutations of objects where some are identical, always divide by the factorial of the number of identical objects to avoid overcounting.