The number of functions $f$ from the set A $=\{0,1,2\}$ in to the set $B=\{0,1,2,3,4,5,6,7\}$ such that $f\left(i\right)\le f\left(j\right)\text{ for }i<j$ and $i,j\in A$ is 

A function $f:A\to B$ such that $f\left(0\right)\le f\left(1\right)\le$$f\left(2\right)$ falls in one of the following four categories.

Case 1  $f\left(0\right)<f\left(1\right)<f\left(2\right)$

There are ${ }^8{C}_3$ functions in this category.

Case 2 $f\left(0\right)=f\left(1\right)<f\left(2\right)$

There are ${ }^8{C}_2$ functions in this category

Case 3    $f\left(0\right)<f\left(1\right)=f\left(2\right)$

There are again ${ }^8{C}_2$ functions in this category

Case 4   $f\left(0\right)=f\left(1\right)=f\left(2\right)$

There are ${ }^8{C}_1$ functions in this category. Thus, the number of desired functions is

${ }^8{C}_3{+}^8{C}_2{+}^8{C}_2{+}^8{C}_1{=}^9{C}_3{+}^9{C}_2{=}^{10}{C}_3$.