Question

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to ______.

Answer 1

We are asked to find the sum of all four-digit numbers that can be formed using the digits 2, 1, 2, and 3. 

Step 1: Total number of distinct four-digit numbers 
The total number of distinct four-digit numbers that can be formed using the digits 2, 1, 2, and 3 is given by the formula for permutations of multiset: \[ \frac{4!}{2!} = \frac{24}{2} = 12. \] This is because the digit '2' is repeated twice, so we divide by \( 2! \) to account for the repetition. Therefore, there are 12 distinct numbers. 

Step 2: Sum of all the numbers 
The four digits that can be used to form the numbers are 2, 1, 2, and 3. The sum of the numbers can be computed by considering the contribution of each digit to the place values (thousands, hundreds, tens, and units). First, note that each of the digits 2, 1, and 3 will appear in each place value (thousands, hundreds, tens, and units) the same number of times, because the numbers are formed by permutations of the digits 2, 1, 2, and 3. Since there are 12 distinct numbers, and there are 4 positions (thousands, hundreds, tens, and units), each digit will appear \( \frac{12}{4} = 3 \) times in each position. 

Step 3: Calculate the contribution of each digit 
The contribution of each digit to the total sum can be calculated by multiplying the digit by the sum of the place values (1000, 100, 10, 1), and then multiplying by how many times the digit appears in each position (3 times). The total contribution of each digit is: \[ 3 \times (1000 + 100 + 10 + 1) = 3 \times 1111 = 3333. \] Now, calculate the sum of the contributions from each digit: - For the digit 2: \[ 2 \times 3333 = 6666. \] - For the digit 1: \[ 1 \times 3333 = 3333. \] - For the digit 3: \[ 3 \times 3333 = 9999. \] The total sum of all the numbers is: \[ 6666 + 3333 + 9999 = 20000. \] However, since the digit '2' appears twice in the list of digits, we have an additional contribution of: \[ 6666. \] Therefore, the total sum is: \[ 20000 + 6666 = 26664. \] 

Final Answer: The sum of all the four-digit numbers that can be formed using the digits 2, 1, 2, and 3 is \( \boxed{26664} \).