How Many Distinct 4-Digit Numbers Can Be Formed from the Digits 1, 2, 2, and 3?

In this article, we will explore how many distinct 4-digit numbers can be formed using the digits 1, 2, 2, and 3. We will use the formula for permutations of a multiset to find the answer. Let's delve into the details.

Using the Formula for Permutations of a Multiset

The formula for the number of distinct permutations of n items where there are groups of indistinguishable items is given by:

(frac{n!}{n_1! cdot n_2! cdot ldots cdot n_k!})

Here, n is the total number of digits, and n_1, n_2, ldots, n_k are the counts of the indistinguishable items.

Calculating the Total Number of Distinct 4-Digit Numbers

For the digits 1, 2, 2, and 3, we have the following counts:

n 4 (total digits) n_1 1 (for digit 1) n_2 2 (for digits 2) n_3 1 (for digit 3)

Plugging these values into the formula, we get:

(frac{4!}{1! cdot 2! cdot 1!} frac{24}{1 cdot 2 cdot 1} 12)

Therefore, the total number of distinct 4-digit numbers that can be formed using the digits 1, 2, 2, and 3 is 12.

Exploring Further with Permutations

Let's look at how this calculation aligns with the example of 1123, which permutes to 12 integers. Similar calculations can be applied to other permutations:

1124 permutes to 12 integers (same as 1123). 1134 permutes to 12 integers (similar to others). 1234 permutes to 24 integers (twice as many).

The total count of permutations for these combinations is as follows:

{1 1 2 2} – 36 18 four-digit integers. {1 1 3 3} – 36 18 four-digit integers.

All together, the total is 54 four-digit integers.

Considering Repetition

Assuming repetition is allowed, the number of distinct 4-digit numbers that can be formed using the digits 1, 2, and 3 is:

Each place in the four-digit number can be filled by 3 options (1, 2, or 3).

Thus, the total number of four-digit numbers is:

3 × 3 × 3 × 3 81

Therefore, 81 distinct four-digit numbers can be formed when repetition of digits is allowed.

Final Considerations

An interesting variation is when we can use a maximum of 2 each of the three digits in each four-digit number. Let's explore this further:

1122: 6 distinct numbers 1123: 12 distinct numbers 1133: 6 distinct numbers 1223: 12 distinct numbers 1233: ...

For each of the above combinations, the number of distinct four-digit numbers is calculated accordingly.

Thank you to Lee Ohringer for bringing this nuance to the discussion and correcting us. Exploring these variations demonstrates the complexity and depth of the problem, contributing to a comprehensive understanding of permutations and multisets in the realm of 4-digit numbers.