An instrument Flight Instructor failed to verify that his student had properly briefed the LAS North Tower 2 SID as cleared and so after takeoff the student flew the Boulder City 2 SID.

Narrative

Late night departure from VGT. Long day of flying; including broken fuel at Mojave; unplanned reposition to WJF; then an overfill of fuel...searching for a pump; defuel; one of those days. Running late at VGT; student briefed departure. I followed by looking at the plate clipped in his yoke. ATC clearance; for North Town 2 Departure. After departure; approaching LAS; ATC asks us to maintain our own terrain separations. Ask why. He says 'you totally blew that departure.' I then look at the plate; ripped it off the yoke. We had flown the Boulder City 2 Departure instead of the cleared North Town 2. Apologized profusely to ATC; but the facts remain: Gross navigational deviation. I am the Training Manager of this program. I am mandating from now on everyone have their own plates; and a gross error check of both plates before flying a SID. Also; fatigue was probably a factor. Fatigue led to complacency; which lead to what happened. After such a long day; rushing to get home was probably not the best idea. Maybe we should have stayed in VGT. Also: Never trust initial IFR students to have the correct plate at an unfamiliar airport!