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We are tasked with finding how many 4-digit numbers can satisfy the following conditions:

1. All four digits are different.
2. The digits are in descending order, meaning the unit digit is the smallest.
3. The sum of two of the digits equals the sum of the other two digits.

### Step-by-Step Breakdown:

1. **Condition 1: All digits are different.**
   - The digits must be distinct.
   - Since we are dealing with a 4-digit number, the possible digits are from 0 to 9.

2. **Condition 2: The digits are in descending order.**
   - If the digits are \( a > b > c > d \), we know the number is structured such that the first digit is the largest, and the last (units) digit is the smallest.

3. **Condition 3: The sum of two digits is equal to the sum of the other two digits.**
   - This implies there are combinations \( a + d = b + c \) that must hold true for the selected four digits \( a > b > c > d \).

### Approach:
To satisfy these conditions, we need to systematically search through the valid combinations of digits that meet the criteria of distinct digits in descending order and where the sum of two digits equals the sum of the other two digits. 

Let me calculate the valid 4-digit numbers that satisfy these conditions.There are **50** valid 4-digit numbers that satisfy the given conditions:

1. All four digits are different.
2. The digits are in descending order.
3. The sum of two digits equals the sum of the other two digits.

Some examples of valid numbers are:
- (3, 2, 1, 0)
- (7, 6, 5, 4)
- (9, 8, 6, 5)
- (9, 8, 7, 6)

Would you like a deeper explanation or further details on this?

Here are five follow-up questions for further exploration:
1. How would the solution change if repeated digits were allowed?
2. What is the probability of randomly selecting a valid number from all 4-digit combinations?
3. How can this approach be generalized for 5-digit numbers with similar conditions?
4. Can a formula be derived to directly compute the number of valid combinations?
5. How would the constraints change if we allowed ascending orders instead of descending?

**Tip:** When solving problems involving multiple conditions, breaking down each condition individually can simplify complex scenarios.

How many 4-digit numbers can satisfy the following three conditions?
(1) All four digits are different;
(2) The digits are in descending orders (the unit digit is the smallest digit);
(3) The sum of two of the four digits is equal to the sum of the other two digits.

Learn how to calculate how many 4-digit numbers meet the conditions where all digits are different, arranged in descending order, and the sum of two digits equals the sum of the other two. This combinatorics problem involves inequalities and simple Diophantine equations. Suitable for Grades 9-12.

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