# Difference between revisions of "2009 UNCO Math Contest II Problems/Problem 10"

## Problem

Let $S=\left \{1,2,3,\ldots ,n\right \}$. Determine the number of subsets $A$ of $S$ such that $A$ contains at least two elements and such that no two elements of $A$ differ by $1$ when

(a) $n=10$

(b) $n=20$

(c) generalize for any $n$.

## Solution

(a) $133$

(b) $\binom{19}{2}+\binom{18}{3}+\binom{17}{4}+\cdots \binom{11}{10}$

(c) $\binom{n-1}{2}+\binom{n-2}{3}+\binom{n-3}{4}+\cdots$