Consider the set of eight vectors V = a^i + b^j + c^k; a, b, c ∈ {-1, 1}. Three non-coplanar vectors can be chosen from V in 2^p ways. Then p is

Let (1,1,1), (-1,1,1), (1,-1,1), (-1,-1,1) be vectors a, b, c, d.
Rest of the vectors are -a, -b, -c, -d and let us find the number of ways of selecting coplanar vectors.
Observe that out of any 3 coplanar vectors two will be collinear (anti parallel).
Number of ways of selecting the anti parallel pair = 4
Number of ways of selecting the third vector = 6
Total = 24
Number of non coplanar selections = 8C3 - 24 = 56 - 24 = 32 = 2^5, p = 5
Alternate Solution:
Required value = 8 × 6 × 4 / 3! ∴ p = 5