is asking how many ways can I arrange n objects in 2n spots.
So lets break this up into a two part problem. For simplicity I will use n=3 and let you generalize
So we have
So we have 3 objects and 6 six spots
_ _ _ , _ _ _
Now we ask if I put 0 ojects in the first 3 spots I can do this in
ways, but this forces me to put all 3 in the last 3 spots and I can do this in
Now if I put 1 object in the first 3 spots I then have to put 2 in the last 3 I can do this in
Next will be 2 objects in the first 3 spots and 1 in the last three with
and finally 3 objects in the first 3 spots and none in the last three gives
So the total number of ways to do this is
Now remember that
using this we get
This is the logic you need for the general proof.
I hope this helps. Good luck.