Explanation:
it all starts with the last statement :
306 take all 3 courses.
so,
V = 306.
from there we go to the statements covering 2 areas, as these 306 must be part of each.
687 science and math = V + VI = 306 + VI.
VI = 687 - 306 = 381
580 psychology and math = V + IV = 306 + IV.
IV = 580 - 306 = 274
579 psychology and science = V + II = 306 + II.
II = 579 - 306 = 273
from there now we can go to the statements covering only 1 area.
1129 psychology = I + II + IV + V = I + 273 + 274 + 306.
I = 1129 - 273 - 274 - 306 = 276
1301 science = III + II + V + VI = III + 273 + 306 + 381.
III = 1301 - 273 - 306 - 381 = 341
1305 math = VII + IV + V + VI = VII + 274 + 306 + 381.
VII = 1305 - 274 - 306 - 381 = 344
and for the total (students not taking any of the 3 classes)
VIII = 2567 - I - II - III - IV - V - VI - VII =
= 2567 - 276 - 273 - 341 - 274 - 306 - 381 - 344 =
= 372