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| Let's count (Posted on 2010-05-16) |
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How many subsets of (1; 2; 3; ... 14) have 15 as the sum of their largest and smallest elements in the subset?
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Submitted by Ady TZIDON
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Rating: 2.5000 (2 votes)
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Solution:
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(Hide)
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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
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