Question

If $A=\{1,2,3\},B=\{1,2,3,4\}$, 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, $n\left(A\right)=3,n\left(B\right)=4$

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 $n\left(A\right)\le n\left(B\right)$

$\therefore$ $Numberofpossibleone-onefunctionsfromAtoB$.

${}^4{P}_3=\frac{4!}{(4-3)!}$ since $n\left(A\right)\le n\left(B\right)$

$=\frac{4\times 3\times 2\times 1}{1}=24$

$\therefore$Number of possible one-one functions from $A$ to $B$ is $24$