number of onto function from $f$ from $\{1,2,3,4,5,\cdots,20\}$ onto $\{1,2,3,4,5,\cdots,20\}$

Question

The number of onto function from $f$ from $\{1,2,3,4,5,\cdots,20\}$ onto $\{1,2,3,4,5,\cdots,20\}$ such that $f(k)$ is a multiple of $3,$ whenever $k$ is a multiple of $4$ is

what i try

Let $A=\{4,8,12,16,20\}$ and $B=\{3,6,9,12,15,18\}$

so we have to find number of onto function from $A$ to $B$

i have seems that answer is $0$ but answer given as $6!\cdot 15!$

How do i solve it Help me please

Answer 1

Since the function is onto (surjective) and the sets are finite and of equal cardinality, the function must be a bijection.

For each number $a\in A$ we have to choose its image $b\in B$; there are $6×5×4×3×2=6!$ ways to do this. Then there are $15!$ ways to assign the remaining mappings for $f$, yielding $6!×15!$ ways in all.