(b) Let $A=\{V, W, X, Y, Z, v, w, x, y, z\}$. Find the number of subsets of the 2 configuration
$$
\begin{array}{c}
\{\{V, W\},\{W, X\},\{X, Y\},\{Y, Z\},\{Z, V\},\{v, x\},\{v, y\},\{w, y\},\{w, z\},\{x, z\}, \\
\{V, v\},\{W, w\},\{X, x\},\{Y, y\},\{Z, z\}\}
\end{array}
$$
that are consistent of order 1 .
Solution: No more than two of the pairs $\{v, x\},\{v, y\},\{w, y\},\{w, z\},\{x, z\}$ may be included in a 2-configuration of order 1 , since otherwise at least one of $v, w, x, y, z$ would occur more than once. If exactly one is included, say $\{v, x\}$, then $w, y, z$ must be paired with $W, Y, Z$, respectively, and then $V$ and $X$ cannot be paired. So either none or exactly two of the five pairs above must be used. If none, then $v, w, x, y, z$ must be paired with $V, W, X, Y, Z$, respectively, and we have 1 2-configuration arising in this manner. If exactly two are used, we can check that there are 5 ways to do this without duplicating an element:
$$
\{v, x\},\{w, y\} \quad\{v, x\},\{w, z\} \quad\{v, y\},\{w, z\} \quad\{v, y\},\{x, z\} \quad\{w, y\},\{x, z\}
$$

In each case, it is straightforward to check that there is a unique way of pairing up the remaining elements of $A$. So we get 52 -configurations in this way, and the total is 6 .