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Find the number of subsets of {1, 2, 3, ..., 10}, where the sum of the smallest element and the largest element is equal to 11.

 Jun 21, 2021
 #1
avatar+179 
+2

341

 

The solution is ... complicated

 Jun 21, 2021
edited by EnchantedLava68  Jun 21, 2021
 #3
avatar+287 
+2

 

Let's elaborate.  We define an $n$--$x$ set to be a set having $n$ as the minimum element, and $x$ as the maximum element such that $n+x=11$.   E.g., $\{4, 5, 7\}$ is a 4--7 set.  There are five possibilities according to the extreme elements.   

 

(i) 5--6 sets: There's only one set $\{5, 6\}$.

(ii) 4--7 sets: Besides 4 and 7 that need to belong, the numbers 5 and 6 can be chosen to be included or excluded independently.  So there are $2^2$ sets of this kind.

(iii) 3--8 sets: Besides 3 and 8 that need to belong, the numbers 4, 5, 6, and 7 can be chosen to be included or excluded independently.  So there are $2^4$ sets of this kind.

(iv) 2--9 sets: Besides 2 and 9 that need to belong, the numbers 3, 4, 5, 6, 7, and 8 can be chosen to be included or excluded independently.  So there are $2^6$ sets of this kind.

(v) 1--10 sets: Besides 1 and 10 that need to belong, the numbers 2, 3, 4, 5, 6, 7, 8, and 9 can be chosen to be included or excluded independently.  So there are $2^8$ sets of this kind.

 

Adding numbers from all five possibliities give the answer.
 

 Jun 21, 2021
 #4
avatar+121048 
0

Very  nice  solution, Bginner  !!!!!

 

 

cool cool cool

CPhill  Jun 22, 2021

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