Let $m$ and $n$ be two non-zero natural numbers:

(a) if $m \leqslant n$, how many strictly increasing functions are there from the set $\{1,2, \ldots, m\}$ to the set $\{1,2, \ldots, n\}$?
(b) in the general case, how many non-decreasing functions are there from the set $\{1,2, \ldots, m\}$ to the set $\{1,2, \ldots, n\}$?
(a) Choosing a strictly increasing function $f:\{1,2, \ldots, m\} \mapsto\{1,2, \ldots, n\}$ is equivalent to choosing the image of $\{1,2, \ldots, m\}$ by $f$, that is, the set $\{f(1), f(2), \ldots, f(m)\}$. Indeed, once $f$ is chosen, this set is defined unambiguously, and conversely, if we know the set $\{f(1), f(2), \ldots, f(m)\}$, we know that $f(k)$ is the $k^{\text{th}}$ smallest element of this set. There are $\binom{n}{m}$ ways to choose such a set: there are therefore $\binom{n}{m}$ functions $f:\{1,2, \ldots, m\} \mapsto\{1,2, \ldots, n\}$ that are strictly increasing.

(b) A function $f:\{1,2, \ldots, m\} \mapsto\{1,2, \ldots, n\}$ is increasing if and only if the function $g:\{1,2, \ldots, m\} \mapsto\{1,2, \ldots, m+n\}$ defined by $g: x \mapsto f(x)+x$ is strictly increasing: there are as many increasing functions $f:\{1,2, \ldots, m\} \mapsto\{1,2, \ldots, n\}$ as there are strictly increasing functions $g:\{1,2, \ldots, m\} \mapsto\{1,2, \ldots, m+n\}$. There are therefore $\binom{m+n}{m}$ such functions.