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  • Using numbers from  S = $\left \{ 1,2,3,4.......n \right \}$
  • We can use maximum up to m numbers to form a set using numbers from S. Repetition of numbers allowed.
  • How many ways we can form a set such that,  $\sum x_i = K$.  Where $K$ is another positive integer. Where $x_i$ are the elements belong $S$ that are included in the newly formed set.

For example :

  • S = $\left \{ 1,2,3,4,5...11,12 \right \}$
  •  m = $4$
  • if $K = 6$
  • Then possible few possible sets are $\{2,4\}, \;\; \{1,3,2\}, \;\; \{1,4,1\},\;\; \{1,1,1,3\}$ etc.
  • $\{1,1,1,1,2\}$ is not valid set for example.
  • Now how many such sets for a particular instance of the problem ? with 
  • S = $\left \{ 1,2,3,4,5,6...12 \right \}$ , $m = 5$, $K = 8$ ?
  • If there is any generic idea ?
  • Ordered / Unordered both the cases !
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1 Answer

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in my openion no of subset possible with a given set is countable ,

 

approach is we have to enumarate it by some another way

first take 0 length subset which is 1

then take 1 lenth subset which is n

then take 2 lenth subset and count all the two length subset passoibel

similarly we can enumarate them so , its countable

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