How many 4-digit numbers have non-increasing digits?

To find how many 4-digit numbers have non-increasing digits, we need to consider a few key factors.

### Non-Increasing Digits
A number is said to have non-increasing digits if, reading from left to right, each digit is either the same as or smaller than the digit to its left. For example, the number 7551 has non-increasing digits, while 7532 does not.

We are tasked with determining how many 4-digit numbers exhibit this property.

### Approach
The digits of a 4-digit number can range from 0 to 9, but since we're looking for 4-digit numbers, the leading digit cannot be zero. The rest of the digits can include zeros.

- To count the number of valid numbers, this becomes a problem of choosing 4 digits from the set {0, 1, 2, ..., 9}, allowing repetition, and arranging them in a non-increasing order.
- This is equivalent to a combinatorial problem, where we are choosing 4 digits from 10 digits (0 through 9), with non-increasing order.

### Combinatorial Setup
This problem can be transformed into a "stars and bars" problem by considering the 10 digits and counting how many of each digit is used. We are looking for solutions to the equation:

\[
x_0 + x_1 + x_2 + \cdots + x_9 = 4
\]

where each \(x_i\) represents the number of times digit \(i\) appears, and \(x_0\) represents how many times 0 appears. Since the number must be a 4-digit number, the count of \(x_0\) must be less than 4 (it can't have all zeros).

The number of ways to choose non-increasing digits without restrictions is:

\[
\binom{4 + 10 - 1}{4} = \binom{13}{4} = 715
\]

However, we must subtract the cases where the number contains only zeros, i.e., \(x_0 = 4\). There is only one such case: 0000.

Thus, the total number of 4-digit numbers with non-increasing digits is:

\[
715 - 1 = 714
\]

### Conclusion
There are **714** four-digit numbers with non-increasing digits.

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Would you like more details or have any questions on this?

Here are five related questions:
1. How many 3-digit numbers have non-increasing digits?
2. How would the solution change for non-decreasing digits?
3. What is the number of 5-digit numbers with non-increasing digits?
4. How can we approach this problem for numbers with exactly 6 digits?
5. What if we want to find numbers where the digits are strictly decreasing?

**Tip:** Stars and bars is a useful combinatorial technique for distributing identical items (stars) into distinct groups (bars).

Learn how to determine how many 4-digit numbers have non-increasing digits using combinatorial techniques like stars and bars. The total number of valid numbers is 714, derived through a combination formula and subtraction of invalid cases. Suitable for students in Grades 10-12.

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