Question

If $A=\{a,b,c\},B=\{1,2,3\}$, then find the number of possible one-one functions from $A$ to $B$.

Answer 1

One-one functions are possible from $A$ to $B$ when $n\left(A\right)\le n\left(B\right)$

Here, $n\left(A\right)=3,n\left(B\right)=3$

Number of possible one-one functions from $A$ to $B$.

${}^{n\left(B\right)}{P}_{n\left(A\right)}=\frac{n\left(B\right)!}{(n\left(B\right)-n\left(A\right))!}$ if $n\left(A\right)\le n\left(B\right)$

$\therefore$Number of possible one-one functions from $A$ to $B$.

${}^3{P}_3=\frac{3!}{(3-3)!}$ since $n\left(A\right)=n\left(B\right)$

$=\frac{3\times 2\times 1}{1}=6$

$\therefore$Number of possible one-one functions from $A$ to $B$ is $6$