Let $A=\{1,2,…,6\} , B=\{1,2,…,20\}$. How many functions $f: B\to B$ are there, such that $I_A \subseteq f$ and $f$ is onto?

Question

Let $A=\{1,2,…,6\} , B=\{1,2,…,20\}$. How many functions $f: B\to B$ are there, such that $I_A \subseteq f$ and $f$ is onto?

I know that there are $|B|^{|B|}$ options for functions, meaning ${20^{20}}$. Usually when I need to tell how many onto functions there are, I'm using the inclusion–exclusion principle, but I also know I need to take into consideration that $I_A \subseteq f$ and $|A| = 6.$ I do know that I will need to subtract the $6$ "bad options" when $I_A \subsetneq f$ but I'm not sure how to go from there. Any help is greatly appreciated.

Answer 1

For $f$ to be onto, $f$ needs to be one-to-one as well, as it is mapping the set to a same-sized set. Such functions must therefore map $1,2,3,4,5,6$ to $1,2,3,4,5,6$, respectively, and permute $7,8,\ldots,20$. Thus the number of such functions is $14!$.