(c) Let $A=\left\{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{10}, b_{10}\right\}$, and consider the 2-configuration $C$ consisting of $\left\{a_{i}, b_{i}\right\}$ for all $1 \leq i \leq 10,\left\{a_{i}, a_{i+1}\right\}$ for all $1 \leq i \leq 9$, and $\left\{b_{i}, b_{i+1}\right\}$ for all $1 \leq i \leq 9$. Find the number of subsets of $C$ that are consistent of order 1 .
Solution: Let $A_{n}=\left\{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{n}, b_{n}\right\}$ for $n \geq 1$, and consider the 2-configuration $C_{n}$ consisting of $\left\{a_{i}, b_{i}\right\}$ for all $1 \leq i \leq n,\left\{a_{i}, a_{i+1}\right\}$ for all $1 \leq i \leq n-1$, and $\left\{b_{i}, b_{i+1}\right\}$ for all $1 \leq i \leq n-1$. Let $N_{n}$ be the number of subsets of $C_{n}$ that are consistent of order 1 (call these "matchings" of $C_{n}$ ). Consider any matching of $C_{n+2}$. Either $a_{n+2}$ is paired with $b_{n+2}$, in which case the remaining elements of our matching form a matching of $C_{n+1}$; or $a_{n+2}$ is paired with $a_{n+1}$, in which case $b_{n+2}$ must be paired with $b_{n+1}$, and the remaining elements form a matching of $C_{n}$. It follows that $N_{n+2}=N_{n+1}+N_{n}$. By direct calculation, $N_{1}=1$ and $N_{2}=2$, and now computing successive values of $N_{n}$ using the recurrence yields $N_{10}=89$.

Define a $k$-configuration of $A$ to be $m$-separable if we can label each element of $A$ with an integer from 1 to $m$ (inclusive) so that there is no element $E$ of the $k$-configuration all of whose elements are assigned the same integer. If $C$ is any subset of $A$, then $C$ is $m$-separable if we can assign an integer from 1 to $m$ to each element of $C$ so that there is no element $E$ of the $k$-configuration such that $E \subseteq C$ and all elements of $E$ are assigned the same integer.