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

If the function is one one, then select any three from the set B in ${ }^7{\mathrm{C}}_3$ ways i.e. 35 ways.
If the function is many one, then there are two possibilities. All three corresponds to same element
number of such functions = ${=}^7{\mathrm{C}}_1=7\text{ ways. }$
Two corresponds to same element, Select any two from the set A The larger one corresponds to the larger and
the smaller one corresponds to the smaller the third `may corresponds to any two.
Number of such functions ${=}^7{\mathrm{C}}_2\times 2=42$
So, the required number of mappings = 35 + 7 + 42 =84