Find the number of maps $f:\{1,2,3\}\to \{1,2,3,4,5\}$ such that $f(i)\le f(j)$ whenever $i<j$.

We are asked to find the number of non-decreasing maps $f:\{1,2,3\}\to \{1,2,3,4,5\}$, i.e., maps such that $f(1)\le f(2)\le f(3)$.

This is equivalent to choosing $3$ elements from the set $\{1,2,3,4,5\}$ with repetition allowed, and then arranging them in non-decreasing order (which can be done in exactly one way).

Using the combination with repetition formula (Stars and Bars):

where $n=5$ (size of the codomain) and $k=3$ (size of the domain).

Therefore, the number of such maps is $35$.