How Many Unique Four-Digit Combinations Can You Make with 1123?

To determine the number of unique four-digit combinations that can be formed from the digits 1, 1, 2, and 3, we need to carefully follow several steps. Let's break it down:

Step 1: Identify the Digits

Our digits are 1, 1, 2, and 3. Note that there are two 1's, one 2, and one 3. This is significant because we need to account for the repetition of the digit 1.

Step 2: Calculate the Total Permutations

The formula for calculating permutations of a multiset (where elements might repeat) is given by:

Permutations frac{n!}{n_1! times n_2! times ... times n_k!}

Where:

n is the total number of items (digits), which in our case is 4. n_1, n_2, ..., n_k are the counts of each distinct item.

For the digits 1, 1, 2, 3:

n_1 (for the digit 1) 2 n_2 (for the digit 2) 1 n_3 (for the digit 3) 1

Substituting these into the formula:

Permutations frac{4!}{2! times 1! times 1!}

Calculating the factorials:

4! 4 times 3 times 2 times 1 24

2! 2 times 1 2

1! 1

Permutations frac{24}{2 times 1 times 1} frac{24}{2} 12

Therefore, there are 12 unique four-digit combinations that can be formed with the digits 1, 1, 2, and 3.

Step 3: Verify the Permutations

Let's list and verify these combinations manually:

1123 1132 1213 1231 1312 1321 2113 2131 2311 3112 3121 3211

As we can see, there are indeed 12 unique combinations.

Conclusion

In summary, there are 12 unique four-digit combinations that can be made with the digits 1, 1, 2, and 3. This calculation is based on the permutation formula for multisets and is verified by manual enumeration.