Please give a detailed explanation for each sub$-$question

$(15$ points$)$

$($a$)$ Suppose $|A|=4$ and $|B|=10.$ Find the number of functions ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$:$AB.$

$($b$)$ Suppose $|A|=4$ and $|B|=10.$ Find the number of $1-1$ functions $f$:$AB.$

$($c$)$ Suppose $|A|=10$ and $|B|=4.$ Find the number of $1-1$ functions $f$:$AB.$

Solution: