Question: The number of maps $f$ from the set $\{1,2,3\}$ into the set $\{1,2,3,4,5\}$ such that $f(i)\le f(j)$, whenever $i<j$.

Answer: 
You have to choose the $3$ function values $f(1),f(2),f(3)$ from $5$ possibilities, with repetition allowed. The order of the $3$ chosen numbers is not important, because once you have chosen them, the condition $f(i)\le f(j)$ tells you which is $f(1)$, which is $f(2)$ and which is $f(3)$.

The number of choices of $3$ elements from $5$, with order not important and repetition allowed, is
$$C(5+3-1,3)=C(7,3)=35\ .$$
For the ideas behind this see [here](https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29).
Hint - Suppose you map $1$ to $1$, now if $2$ maps to $1$ then $3$ can map to $1,2,3,4$ or $5$, Similarly if $2$ maps to $2$ then $3$ will have $4$ choices to map to, so the total no of functions when $1$ maps to $1$ is 

$(5+4+3+2+1)$

Similarly count the others when $1$ maps to $2,3,4$ or $5$