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How many subsets? (Posted on 2015-06-29) Difficulty: 3 of 5
Given that S = {1,2,3,.....,14}, determine the total number of 7-element subsets of S such that the sum of elements of each of the subsets is divisible by 14

No Solution Yet Submitted by K Sengupta    
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Solution simple count | Comment 1 of 3

I believe the answer is 1+21+35+7= 64.

1 for the basic set of 1,2,3...7 - sums up to 28.

COMB(7,2)=21  for sets formed by augmenting any 2 of the 7 numbers by 7, like  8,9,3,4,5,6,7

COMB(7,4)=35  for sets formed by augmenting any 4 of the 7 numbers by 7, like  8,9,10,11,5,6,7

COMB(7,6)=7  for sets formed by augmenting any 6 of the 7 numbers by 7, like  8,9,10,11,12,13,7

0,2,4 or 6 members of (1,2,3,4,5,6,7) needed to be changed- half of the sum of the 7th line in Pascal triangle, i.e. 2^6=64


Edited on June 29, 2015, 4:38 pm
  Posted by Ady TZIDON on 2015-06-29 16:24:32

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