The problem asks for the number of 4-digit numbers that satisfy the following three conditions:

1. All four digits are different.
2. The digits are in descending order (the unit digit is the smallest).
3. The sum of two of the four digits is equal to the sum of the other two digits.

Let’s break down the solution:

Step 1: Understanding the Conditions

• Condition 1: The four digits must be distinct, which implies no repetitions of digits.
• Condition 2: The digits must be in strictly descending order, so we can choose any four distinct digits and then sort them such that $d_1 > d_2 > d_3 > d_4$, where $d_4$ is the unit digit.
• Condition 3: The sum of two of the digits is equal to the sum of the other two. This adds a restriction, where we need to select four digits that can be partitioned such that their sums match, i.e., $d_1 + d_4 = d_2 + d_3$ (since the order must be descending).

Step 2: Choosing Distinct Digits

There are 10 digits to choose from (0 through 9). However, for a four-digit number, the first digit cannot be 0. So, we must select from the set $\{1, 2, 3, \ldots, 9\}$ for the highest place.

Step 3: Ensuring the Sum Condition

For each combination of four distinct digits, we need to check if it's possible to partition them into two pairs whose sums are equal. This introduces some combinatorial complexity, as not all sets of four digits can satisfy this condition.

Step 4: Counting Valid Combinations

We need to calculate the number of sets of four distinct digits where the descending order and sum condition are met. This involves generating combinations of four digits, arranging them in descending order, and checking if the sum condition holds.

Let me calculate the number of such combinations next.There are 34 valid combinations of four distinct digits that satisfy the given conditions:

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

Some examples of valid combinations are:

• $(1, 2, 3, 4)$
• $(1, 2, 8, 9)$
• $(4, 6, 7, 9)$

These combinations satisfy all the criteria, including the descending order and the sum condition $d_1 + d_4 = d_2 + d_3$.

Do you need more details or any clarifications? Here are some related questions for further practice:

1. How many 4-digit numbers can be formed if there are no restrictions on the sum condition?
2. What is the probability of randomly selecting four digits that satisfy the sum condition?
3. Can this type of sum condition be generalized for 5-digit numbers?
4. How does the solution change if repetition of digits is allowed?
5. What are the properties of the combinations where the sum condition is satisfied?

Tip: When solving problems with multiple conditions, tackle each condition step by step to simplify the process!