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

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== Solution ==
 
== Solution ==
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(a) <math>133</math> 
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(b) <math>\binom{19}{2}+\binom{18}{3}+\binom{17}{4}+\cdots \binom{11}{10}</math> 
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(c) <math>\binom{n-1}{2}+\binom{n-2}{3}+\binom{n-3}{4}+\cdots</math>
  
 
== See also ==
 
== See also ==
{{UNC Math Contest box|year=2009|n=II|num-b=9|num-a=11}}
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{{UNCO Math Contest box|year=2009|n=II|num-b=9|num-a=11}}
  
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]

Latest revision as of 01:59, 13 January 2019

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

See also

2009 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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