To find how many 4-digit numbers have non-increasing digits, we need to count all possible 4-digit numbers where each digit is greater than or equal to the next digit. Let's represent a 4-digit number as $abcd$ where $a \geq b \geq c \geq d$ and each of $a$, $b$, $c$, and $d$ are digits from 0 to 9.

Solution Outline

1. Define the Range for Each Digit: Since we are dealing with 4-digit numbers, $a$ (the thousands place) must be non-zero (i.e., $a \geq 1$), while $b$, $c$, and $d$ can each be any digit from 0 to 9.

2. Apply Combinatorics: We need to find combinations of digits such that they are non-increasing. To do this, we can use a combinatorial approach with the concept of "stars and bars" or "combinations with repetition."

3. Transform the Problem: Let's define a new variable to count how often each digit appears in our number. For instance, let:

   • $x_1$ represent the count of 9s,
   • $x_2$ the count of 8s,
   • $x_3$ the count of 7s,
   • ...
   • $x_{10}$ represent the count of 0s.

   These counts satisfy: $x_1 + x_2 + \cdots + x_{10} = 4$ because the total number of digits in any 4-digit number is 4.

4. Constraints: Ensure that the first non-zero digit appears in the thousands place to avoid 0 starting the number. We can handle this by subtracting combinations that start with zero.

5. Calculate Using Combinations with Repetition: The number of solutions to the equation $x_1 + x_2 + \cdots + x_{10} = 4$ (where each $x_i \geq 0$) is given by the formula for combinations with repetition: $\binom{4 + 10 - 1}{10 - 1} = \binom{13}{9} = 715$

After accounting for restrictions on the first digit (ensuring it’s non-zero), this approach provides the count of all valid non-increasing 4-digit numbers.

Final Answer

There are 715 four-digit numbers with non-increasing digits.

Would you like more details on the steps?

1. How do we handle non-increasing sequences in other lengths, like 5-digit numbers?
2. Can we use similar combinatorial approaches for non-decreasing digits?
3. How do we calculate combinations with constraints on specific digits?
4. Could we explore examples of 4-digit numbers with repeating digits?
5. What’s the role of “stars and bars” in combinatorial math?

Tip: Non-increasing sequences follow a systematic pattern, making them ideal for combinatorics-based counting.