find the sum of all the four digits number that can be formed with the digit 0235

To find the sum of all the four-digit numbers that can be formed using the digits 0, 2, 3, and 5, we need to consider the permutations of these digits. Here's a step-by-step breakdown of the solution:

1. Break Down the Problem

   • Identify all possible four-digit numbers that can be formed using the digits 0, 2, 3, and 5.
   • Ensure that the number does not start with 0, as it would not be a valid four-digit number.

2. Relevant Concepts

   • Permutation: The number of ways to arrange a set of items.
   • For four distinct digits, the number of permutations is $4!$ (factorial of 4).
   • However, since 0 cannot be the leading digit, we need to adjust our calculations accordingly.

3. Analysis and Detail

   • Calculate the total permutations: $4! = 24$.

   • Exclude permutations where 0 is the leading digit. If 0 is the leading digit, the remaining three digits (2, 3, 5) can be arranged in $3!$ ways.

   • Therefore, the valid permutations are $24 - 6 = 18$.

   • Calculate the contribution of each digit in each position (thousands, hundreds, tens, units):

     • Each digit appears in each position an equal number of times across all valid permutations.
     • For example, in the thousands place, each of the digits 2, 3, and 5 appears $\frac{18}{3} = 6$ times.

   • Sum of digits: $0 + 2 + 3 + 5 = 10$.

4. Verify and Summarize

   • Calculate the total sum by considering the contribution of each digit in each position:

     • Thousands place: $6 \times (2 + 3 + 5) \times 1000 = 60,000$
     • Hundreds place: $6 \times (0 + 3 + 5) \times 100 = 4,800$
     • Tens place: $6 \times (0 + 2 + 5) \times 10 = 420$
     • Units place: $6 \times (0 + 2 + 3) \times 1 = 30$

   • Total sum: $60,000 + 4,800 + 420 + 30 = 65,250$.

Final Answer

The sum of all the four-digit numbers that can be formed with the digits 0, 2, 3, and 5 is 65,250.