Question

Let $A=\{1,2,3,4,5,6\}$ and $B=\{a,b,c,d\}$, then the total number of possible functions from $A$ to $B$

• A. without any restriction is $6^4$
• B. such that every element in $B$ has atleast one pre-image is $1560$
• C. such that all the elements in $B$ should not have pre-image is $2536$
• D. without any restriction is $4^6$

Answer 1

Answer: (B), (C), (D)

The correct options are
B such that every element in $B$ has atleast one pre-image is $1560$
C such that all the elements in $B$ should not have pre-image is $2536$
D without any restriction is $4^6$

Here, $A=\{1,2,3,4,5,6\}$ and $B=\{a,b,c,d\}$, $n=6,r=4$ then the total number of possible functions from $A$ to $B$ is $r^n=4^6...\left(1\right)$

Total number of possible functions from $A$ to $B$ such that every element in $B$ has atleast one pre-image is $=r^n-{}^r{C}_1(r-1{)}^n+{}^r{C}_2(r-2{)}^n+...+(-1{)}^{r-1}\cdot {}^r{C}_{r-1}(1{)}^n$
$=4^6-{}^4{C}_1(3{)}^6+{}^4{C}_2(2{)}^6-{}^4{C}_3(1{)}^6=1560...\left(2\right)$

Total number of possible functions from $A$ to $B$ such that all the elements in $B$ should not have pre-image $=\left(1\right)-\left(2\right)$
$4^6-1560=2536$