Question

Number of onto (surjective) functions from $A$ to $B$ if $n(A)=6$ and $n(B)=3$ are

• A. $2^6-2$
• B. $3^6-3$
• C. $340$
• D. $540$

Answer 1

Answer: (D)

Number of onto functions from $A$ to $B$ if $n(A)=m$
$n(B)=n$ and $1\le n\le m$ are equal to
$\sum _{r=1}^n(-1{)}^{n-r}{}^n{C}_r{r}^m$
Here $n=3,m=6$
$\therefore$ Number of onto functions
$=\sum _{r=1}^3(-1{)}^{3-r}{}^3{C}_r{r}^6$
$=(-1{)}^3{}^3{C}_1{1}^6+(-1{)}^1{}^3{C}_2{2}^6+(-1{)}^0{}^3{C}_3{3}^6$
$=3-3\times 2^6+3^6$
$=3(3^5-2^6+1)=540$