Suppose wanting to count the amount of strings of length n that are non-decreasing.

eg: abc, aaa, abb qualify; cba doesn't.

Well, well, well. This is a problem very much akin to what Project Euler usually sets up, in the sense that there is a possible programming approach (one that I tried, naively at first, until realising this is much better accomplished with some dynamic programming (with the help of Phind):

def count_non_decreasing_strings(n):
    # Initialize a 2D list to store the counts of non-decreasing strings ending with each character
    dp = [[0 for _ in range(n+1)] for _ in range(26)] # Assuming lowercase English letters
    
    # Base case: a single character string is always non-decreasing
    for i in range(26):
        dp[i][1] = 1
    
    # Fill the table in a bottom-up manner
    for length in range(2, n+1):
        for last_char in range(26):
            # Sum of all strings of length length-1 ending with a character <= last_char
            for prev_char in range(last_char+1):
                dp[last_char][length] += dp[prev_char][length-1]
    
    # The total count of non-decreasing strings of length n is the sum of all counts for the last character
    total_count = sum(dp[i][n] for i in range(26))
    return total_count

...but, of course, there is a much simpler, quicker solution, relying on the mathematics of combinatorics.

Mathematics establishes the concept of a set, which is an unordered assortment of elements that do not repeat (eg. $\{1, 3, 2\}$ is a set — the same set as $\{1, 2, 3\}$ — but $\{1, 2, 2\}$ is not a set); but there's also the multiset, which allows for repetition (where $\{1, 2, 2\}$ is now a valid multiset).

With this in mind, this problem can be modelled by considering combinations with repetitions. Let $S$ be a set of $26$ elements — think of abcd...xyz — from which one can take $n$ elements in a sequence, putting back (or possibly taking multiple) of each element. Thus, abab is valid for $n = 4$.

The hardest part of modelling in this way, I suppose, is wrangling the utility of a multiset (unordered by nature) in representing our ordered strings. See it this way: retrieve all multisets of $3$ letters: $\{a, b, c\}$ is there, and so is $\{b, c, c\}$ — all good so far — but $\{z, d, c\}$ is also there. But a multiset is unordered, and there is only one non-decreasing string with $\{z, d, c\}$ as elements: cdz. Same thought process for $\{b, c, c, a, b\}$: abbcc. So each multiset maps to exactly one non-decreasing string, transforming the counting of strings into the counting of possible multisets...which is given by $n$ multichoose $k$, in the following binomial $\binom{26 + n - 1}{26}$