If set $A$ contains $8$ distinct numbers, and set $B$ contains $5$ distinct numbers, and $f : A \to B$ is a function, then prove that the number of non-decreasing functions possible from $A$ to $B$ is $495$.
In the solutions the final answer had been given as $\binom{12}{4}$ suggesting that the answer arose from a possible logical series developed for a general case.
Any insight on how to approach these kind of problems will be very helpful.
These non-decreasing functions are in bijection with the arrangements of $8$ balls in $5$ (possibly empty) bins. (Map as many elements of $A$ to an element of $B$ as are balls in the corresponding bin.) By the method of stars and bars the number of arrangements of $n$ balls in $k$ bins can be shown to be $\binom{n+k-1}{k-1}$. In the present case, $n=8$ and $k=5$, so the number of these functions is $\binom{12}4=495$.