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Let's count (Posted on 2010-05-16) Difficulty: 3 of 5
How many subsets of (1; 2; 3; ... 14) have 15 as the sum of their largest and smallest elements in the subset?

  Submitted by Ady TZIDON    
Rating: 2.5000 (2 votes)
Solution: (Hide)
Answer: 5461

If a is the smallest element of such a set, then 15 - a is the largest element,
and for the remaining elements we may choose any (or none) of the 14 - 2a elements: a + 1; a + 2; ... ; (13- a) - 1.

Thus there are 2^(14-2a) sets whose smallest element is a.
Since (15-a)>a causes a < 8, the summation of 2^(14-2a) over l a = 1; 2; ...; 7 provides an answer:
2^12*2^10+2^8+2^6+2^4+2^2+2^0
=4096+1024+256+64+16+4+1=5461



Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: solution (spoiler)- decimal spoilerAdy TZIDON2010-05-16 13:58:09
Solutionsolution (spoiler)Charlie2010-05-16 12:45:42
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