2006 AMC 12A Problems/Problem 25

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Problem

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 $k$.

$\mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{(D) \ } 377$$\mathrm{(E) \ }  405$

Solution

See also