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Let S = {1, 2, …, n} and let T be the set of all ordered triples of subsets, say , such that . Determine, in terms of n, , where |X| denotes the number of elements in the set X. (For example, if S={1, 2, 3} and then one of the elements of T is ({1,2}, {2,3}. {3}).)

  1. Let S = {1, 2, …, n} and let T be the set of all ordered triples of subsets, say (A_1, A_2, A_3)   , such that A_1 \cup A_2 \cup A_3 = S  . Determine, in terms of n, \sum_{(A_1, A_2, A_3) \in T} |A_1 \cap A_2 \cap A_3 |  , where |X| denotes the number of elements in the set X. (For example, if S={1, 2, 3} and A_1=\{1,2\}, A_2=\{2, 3\}, A_3 = \{3\}  then one of the elements of T is ({1,2}, {2,3}. {3}).)

Grade:11

1 Answers

mycroft holmes
272 Points
5 years ago
Suppose we count all triplets A1, A2, A3 that have exactly r elements in common.
 
The number of ways such triplets can be formed is counted as:
(i) we can choose the r>0 elements to be chosen in nCr ways. Now, the remaining (n-r) elements have each the following choices
   (a) they can be part of exactly one set in 3 ways
   (b) they can be part of exactly two sets in 3 ways
  making available 6 choices for each element
Hence number of such triplets is nCr 6n-r. for r>0
 
So the required sum is 
 
This is a regular sum that can be obtained by differentiating (x+6)n and putting x=1

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