For most species, you can include up to 5 choices, but the 4th and 5th choices do not pertain to regular hunt choices, rather they apply to left-over tags. If you are not willing to accept any tag, including antlerless tags, do not include a 4th and 5th choice on your application.
New Mexico considers all 3 of each applicant's choices before moving to the next application. This means that once your application is considered they will attempt to award your first choice if that hunt is filled they move to the next choice and so on. Therefore, the best strategy is to list the better hunts (tougher odds) as your first choice and list progressively better odds hunts as you 2nd and 3rd choice.