I'm attaching a TV wall mount which is shown in the image to external studs that I will install in front of the wall.
Does it matter for the holding strength which holes on the TV mount I use, those at the ends or those closer to the center?
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Based on the picture you provided, I would suggest that, if possible, you should use holes that will allow your lag bolts to go through the horizontal band of wood and the vertical post portion, as well. This will give your bolts the maximum about of solid wood to grip into.
It's hard to tell from that image, but I would suspect that you used 1 by material for the horizontal and 2 by material for the vertical. That would give you a maximum of 2-1/4" of wood for your lags to hold in, where the mount is expecting to be attached to the narrow edge of a 2 x 4 stud, thus allow for 3-1/2" of wood. Since you'll be using shorter lags, you might want to consider adding an extra couple, just to distribute the weight a little more.