Question

Number of onto (surjective ) functions from $A$ to $B$ if $n\left(A\right)=6$ and $n\left(B\right)=3$ is

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

Answer 1

The correct option is D $540$

Number of onto functions from $A$ to $B$ if $n\left(A\right)=m,n\left(B\right)=n$ 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 $=\sum _{r=1}^3(-1{)}^{3-r}{}^3{C}_r{r}^6=(-1{)}^2{}^3{C}_1{1}^6+(-1{)}^1{}^3{C}_2{2}^6+(-1{)}^0{}^3{C}_3{3}^6$ $=(3{)}^6-3\times 2^6+3$ $=3\left(\right(3{)}^5-2^6+1)=540$
Ans: D