Question

Given a function $f:A\to B$; where $A=\{1,2,3,4,5\}$ and $B=\{6,7,8\}$.

The number of mappings of $g\left(x\right):B\to A$ such that $g\left(i\right)\le g\left(j\right)$ whenever $i<j$ is

• A. $55$
• B. $140$
• C. $10$
• D. $35$

Answer 1

The correct option is D $35$

In the first case, let $g\left(8\right)=5$

If $g\left(7\right)=5$, then $g\left(6\right)\in \{1,2,3,4,5\}$

Therefore total number of mapping $=5$

Similarly, If $g\left(7\right)=4$, then $g\left(6\right)\in \{1,2,3,4\}$

Therefore number of mappings $=4$

If $g\left(7\right)=3$, then $g\left(6\right)\in \{1,2,3\}$

Therefore number of mappings $=3$

If $g\left(7\right)=2$, then $g\left(6\right)\in \{1,2\}$

Therefore number of mappings $=2$

If $g\left(7\right)=1$, then $g\left(6\right)=1$

Therefore number of mappings $=1$

Total number of mappings in first case $=5+4+3+2+1=15$

In the second case, Let $g\left(8\right)=4$

If $g\left(7\right)=4$, then $g\left(6\right)\in \{1,2,3,4\}$

Therefore number of mapping $=4$

If $g\left(7\right)=3$, then $g\left(6\right)\in \{1,2,3\}$

Therefore number of mapping $=3$

If $g\left(7\right)=2$, then $g\left(6\right)\in \{1,2\}$

Therefore number of mapping $=2$

If $g\left(7\right)=1$, then $g\left(6\right)=1$

Therefore number of mapping $=1$

The total number of mappings $=4+3+2+1=10$

Similarly, In other cases when $g\left(8\right)=3$, total number of mappings$=3+2+1=6$

$g\left(8\right)=2$, then the number of mappings$=2+1=3$

and $g\left(8\right)=1$, then number of mppings $=1$

Total number of mappings$=15+10+6+3+1=35$