The first question is meant to be "nobody will get his own coat or his own hat"

You don't answer our sincere questions and moreover you don't think we read English.

I doubt that you can write clear English. So we are left to assume that these men pick a coat at random and then pick a hat at random.

Now because $\sum\limits_{k = 0}^\infty {\dfrac{{{{( - 1)}^k}}}{{k!}}} = \dfrac{1}{e}$

We can use that to model $\mathscr{D}_n=\left\lfloor {\frac{{n!}}{e} + \frac{1}{2}} \right\rfloor \text{ for }n\ge 3$.

Look at this table:

SEE HERE
Let $C$ be the event that none of the $n$ men gets his own Coat. Let $H$ be the event that none of the $n$ men gets his own Hat.

$\mathcal{P}(C\cup H)=\mathcal{P}(C)+\mathcal{P}(H)-\mathcal{P}(C\cap H)$ i.e. Does not get his coat

**or** hat.

Now if you were not so self-important you may have done us the honor of explaining the setup of the question.

If there were two different racks, one for coats the other for hats, then clearly the selections would be independent.

So if we assume Independence how do you finish?

**DON'T ASK!**