Number of non-decreasing functions?
Given a non-decreasing function $f:A\to B$, consider the function $\hat f:A\to\{1,2,\ldots,29\}$ given by
$$\hat f(n)=f(n)+n-1$$
In other words,
$$\hat f(1)=f(1),\ \hat f(2)=f(2)+1,\ \ldots,\ \hat f(10)=f(10)+9$$
Then $f$ is non-decreasing if and only if $\hat f$ is strictly increasing. If you look closely you will see that this transformation of $f$ to $\hat f$ is a bijection from {non-decreasing functions from $A$ to $B$} to {strictly increasing functions from $A$ to $\{1,2,\ldots,29\}$}. I need to rush off now so I can't type the details, but I think they will be clear: let me know if you need me to add details when I get back.
The strictly increasing functions are easy to count, since there is a bijection from those functions to the sets of the same size of the domain taken from a set of the same size as the co-codomain. I.e. given a set of size $10$ taken from the set $\{1,2,\ldots,29\}$ we take $\hat f(1)$ as the smallest member of the set, $\hat f(2)$ as the second-smallest, etc.
These two bijections show that your desired number of non-decreasing functions is
$${29 \choose 10}$$
To make things more clear, here is how to get a non-decreasing function $f:\{1,2,\ldots,10\}\to\{1,2,\ldots,20\}$.
First, choose any $10$ distinct numbers from $\{1,2,\ldots,29\}$. Sort them in increasing order. Then we define $f(1)$ as the smallest of those chosen numbers. We define $f(2)$ as the second-smallest of those chosen numbers minus $1$. (This is guaranteed to be greater than or equal to $f(1)$). We define $f(3)$ as the third-smallest of those chosen numbers minus $2$. (This is guaranteed to be greater than or equal to $f(2)$). ... We define $f(10)$ as the tenth-smallest of those chosen numbers minus $9$. (This is guaranteed to be greater than or equal to $f(9)$).
The opposite of a nondecreasing function is not a decreasing function. For example, the function that maps $1 \mapsto 1, \;2 \mapsto 3, \;3 \mapsto 2$ is neither decreasing, nor nondecreasing.
But your idea can be adapted to calculate the number of nondecreasing functions. You just need to consider the number of ways to choose 10 numbers from 20 with repetitions.