Let $S = \left\{ A = \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0,1\} \text{ and } |A| \in \{-1,1\} \right\}$ where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is ______.

A = 0 1 c 1 a d 1 b e , a,b,c,d,e ∈ {0,1}

$|A| = \det(A) \in \{-1, 1\}$

$\text{To find: } |S| \text{ where } S = \{A : a,b,c,d,e \in \{0,1\},\ \det A = \pm 1\}$

$\det A = 0 - 1(e - d) + c(b - a) = d - e + c(b - a)$

$c = 0:\quad \det A = d - e = \pm 1 \iff (d,e) = (1,0) \text{ or } (0,1)$

$\text{count} = 2 \times 2 \times 2 = 8$

$c = 1:\quad \det A = (d - e) + (b - a) = x + y$

$x = d - e,\qquad y = b - a,\qquad x, y \in \{-1,0,1\}$

$x+y = 1:\ (0,1)\to 2,\ (1,0)\to 2 \ \Rightarrow\ 4$

$x+y = -1:\ (0,-1)\to 2,\ (-1,0)\to 2 \ \Rightarrow\ 4$

$\text{count} = 4 + 4 = 8$

$|S| = 8 + 8 = 16$

$\boxed{|S| = 16}$