For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________

<p>To find $ n(S_5) $, which is the number of subsets of $\{1, 2, 3, 4, 5\}$ with no consecutive numbers, we start by enumerating these subsets.</p> <p>Let's denote the set $\{1, 2, 3, 4, 5\}$ as $A$. The subsets of $A$ that meet the criteria are:</p> <p><p>The empty set: $\{\}$</p></p> <p><p>Single-element sets: $\{1\}$, $\{2\}$, $\{3\}$, $\{4\}$, $\{5\}$</p></p> <p><p>Two-element sets with no consecutive numbers: $\{1, 3\}$, $\{1, 4\}$, $\{1, 5\}$, $\{2, 4\}$, $\{2, 5\}$, $\{3, 5\}$</p></p> <p><p>Three-element set with no consecutive numbers: $\{1, 3, 5\}$</p></p> <p>Counting these subsets, we have:</p> <p><p>1 subset with zero elements</p></p> <p><p>5 subsets with one element</p></p> <p><p>6 subsets with two elements</p></p> <p><p>1 subset with three elements</p></p> <p>Adding these counts, there are $1 + 5 + 6 + 1 = 13$ subsets in total.</p> <p>Thus, $ n(S_5) = 13 $.</p>