Question 25
How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 15\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $S.$
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k.$
$(1)$ No two consecutive integers belong to $S.$
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k.$