Permutation and Combination Question with multiple variables

Hi all,

Currently having issues finding clarification on the following question:

A space shuttle mission consists of a commander, a pilot, 3 engineers, a doctor, a scientist, and a civilian. If there are 7 people from whom the commander and pilot must be chosen, 18 possible engineers, 4 possible doctors, 7 scientists and 3 candidates for the civilian crew member, in how many ways can a crew be formed?

My guess is that each part of the "crew" would have to be calculated, and then added.. I need more assistance with explaining how the question is solved, VS receiving the answer directly.

eg) Engineers = ncR = 18ncr3 Doctors = ncr = 4ncr1 Scientist = 7ncr1

Thanks!