Consider the set of eight vectors  $V=\{a\widehat{i}+b\widehat{j}+c\widehat{k};a,b,c\in \{-1,1\}\}$. 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  $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$ . Rest of the vectors are $-\overrightarrow{a},-\overrightarrow{b},-\overrightarrow{c},-\overrightarrow{d}$ and let us find the number of ways of selecting coplanar vectors.
Observe that out of any 3 coplanar vectors, two will be collinear (antiparallel)
Number of ways of selecting the antiparallel pair  $= 4$
Number of ways of selecting the third vector $=6$
Total $= 24$
Number of non-coplanar selections  $={}^8{C}_3-24=32=2^5$
$\therefore p=5$