$f$ is non-decreasing if $x \lt y$ implies $f(x) \leq f(y)$ and increasing if $x < y$ implies $f(x) < f(y)$.
How many $f: [a]\to [b]$ are nondecreasing?
How many $f: [a] \to [b]$ are strictly increasing?
Where $[a]=\{1,2\ldots a\}$ and $[b]=\{1,2\ldots b\}$
Best Answer
Strictly increasing is easy: we need to choose the $n$ items in $[k]$ that will be the range of our function.