How many four digit numbers can we create from the digits
111223?

To determine how many four-digit numbers can be created using the digits from the set \(\{1, 1, 1, 2, 2, 3\}\), we need to consider different scenarios based on how many of each digit is used.

### Case 1: Three 1's and One Other Digit
- We choose three 1's and one other digit (2 or 3).
- Possible combinations:
  - \(1, 1, 1, 2\)
  - \(1, 1, 1, 3\)

For each combination, we calculate the number of distinct four-digit numbers:
- The number of distinct arrangements of \(1, 1, 1, 2\) is given by \(\frac{4!}{3!1!} = 4\).
- The number of distinct arrangements of \(1, 1, 1, 3\) is also \(4\).

So, for this case, there are \(4 + 4 = 8\) numbers.

### Case 2: Two 1's, One 2, and One 3
- The digits are \(1, 1, 2, 3\).
- The number of distinct arrangements is given by \(\frac{4!}{2!1!1!} = \frac{24}{2} = 12\).

So, for this case, there are \(12\) numbers.

### Case 3: Two 2's and Two 1's
- The digits are \(1, 1, 2, 2\).
- The number of distinct arrangements is given by \(\frac{4!}{2!2!} = \frac{24}{4} = 6\).

So, for this case, there are \(6\) numbers.

### Total Number of Four-Digit Numbers
Adding the numbers from each case:
\[ 8 \, (\text{from case 1}) + 12 \, (\text{from case 2}) + 6 \, (\text{from case 3}) = 26 \]

Thus, the total number of distinct four-digit numbers that can be created from the digits \(\{1, 1, 1, 2, 2, 3\}\) is **26**.

Would you like to explore any of these cases in more detail, or do you have any other questions?

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**Here are some related questions you might find interesting:**

1. How many five-digit numbers can be formed using the same digits?
2. How does the answer change if the digits are 111233?
3. What is the probability of randomly selecting a four-digit number made up of these digits that starts with 1?
4. How many different four-digit numbers can be formed if repetition of digits is allowed?
5. What if we needed to form even four-digit numbers?
6. How would the number change if the digit 0 was also included in the set?
7. Can you generalize the method to find how many four-digit numbers can be formed from any given set of digits?
8. What are the smallest and largest four-digit numbers that can be formed from these digits?

**Tip:** When solving problems involving arrangements of objects, factorials (\(n!\)) are often useful in counting distinct permutations, especially when some objects are identical.

Learn how many distinct four-digit numbers can be created using the digits 111223. This problem involves permutations and combinations, focusing on different cases of digit arrangements and factorials. Suitable for students in Grades 9-12.

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