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

A function $\mathrm{f}:\mathrm{A}\to \mathrm{B}$ such that $\mathrm{f}\left(0\right)\le \mathrm{f}\left(1\right)\le \mathrm{f}\left(2\right)$ falls in one of the following categories.$\begin{array}{l}\mathrm{Case}\left(\mathrm{i}\right):\mathrm{f}\left(0\right)<\mathrm{f}\left(1\right)<\mathrm{f}\left(2\right)=8_{{\mathrm{C}}_3}\\ \begin{array}{l}\mathrm{Case}\left(\mathrm{ii}\right):\mathrm{f}\left(0\right)=\mathrm{f}\left(1\right)<\mathrm{f}\left(2\right)\\ \mathrm{f}\left(0\right)<\mathrm{f}\left(1\right)=\mathrm{f}\left(2\right)\end{array}\}=2\times 8_{{\mathrm{C}}_2}\\ \mathrm{Case}\left(\mathrm{iii}\right):\mathrm{f}\left(0\right)=\mathrm{f}\left(1\right)=\mathrm{f}\left(2\right)\}=8_{{\mathrm{C}}_1}\\ \mathrm{the}\mathrm{number}\mathrm{of}\mathrm{desired}\mathrm{function}\mathrm{is}{8}_{{\mathrm{C}}_3}+8_{{\mathrm{C}}_2}+8_{{\mathrm{C}}_2}+8_{{\mathrm{C}}_1}=9{\mathrm{c}}_3+9_{{\mathrm{C}}_2}={10}_{{\mathrm{C}}_3}\end{array}$