How many ways can 2m objects be paired and assigned to m boxes? In how many ways can $2m$ objects be paired and assigned to $m$ boxes? 
In that post the questioner claims that there are $\frac{(2m)!}{2^m}$ to pair $2m$ and assign them to $m$ boxes.
My attempt :-
First we need to find number of ways to pair $2m$ objects. 
That is $\frac{(2m)!}{2! * (2m-2)!}$ = $\frac{2m * (2m - 1)}{2}$ = $m * (2m - 1)$
After pairing there are $m$ objects left, and there are $m!$ to permute them in $m$ boxes
Thus the answer should be $m! * m * (2m -1)$ ways.
Which is wrong. what i am doing incorrectly, please anyone correct me.  

The number of ways that $2m$ objects can be grouped into $m$ boxes where each box contains 2 objects is:
$$ \binom{2m}{2_{b1}, 2_{b2} ...  2_{bm}} = \frac{(2m)!}{2_{b1}!2_{b2}! ... 2_{bm}!} = \frac{(2m)!}{2^m} $$
where $b1$ is the first box, and $bm$ is the last box.
