Counting strictly increasing and non-decreasing functions $f$ is non-decreasing if $x \lt y$ implies $f(x) \leq f(y)$ and increasing if $x < y$ implies $f(x) < f(y)$. 

  
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*How many $f: [a]\to [b]$ are nondecreasing?
  
*How many $f: [a] \to [b]$ are strictly increasing?

Where $[a]=\{1,2\ldots a\}$ and $[b]=\{1,2\ldots b\}$

Strictly increasing is easy: we need to choose the $n$ items in $[k]$ that will be the range of our function. 
