The number of functions ‘f’ from the set $A=\{0,1,2\}$ into the set $B=\{0,1,2,3,4,5,6,7\}$ such that $f\left(i\right)\le f\left(j\right)$ 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 categories.
 $Case\left(i\right):f\left(0\right)<f\left(1\right)<f\left(2\right)={}^8{C}_3$
 $Case\left(ii\right):\begin{array}{l}f\left(0\right)=f\left(1\right)<f\left(2\right)\\ f\left(0\right)<f\left(1\right)=f\left(2\right)\end{array}\}=2\times {}^8{C}_2$
 $Case\left(iii\right):f\left(0\right)=f\left(1\right)=f\left(2\right)={}^8{C}_1$
the number of desired function is ${ }^8{C}_3+{}^8{C}_2+{}^8{C}_2+{}^8{C}_1={}^9{C}_3+{}^9{C}_2={}^{10}{C}_3$