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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?

See The Solution Submitted by Ady TZIDON    
Rating: 2.5000 (2 votes)

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Solution solution (spoiler) | Comment 1 of 2

There are seven possibilities for lowest and highest elements: 1 & 14; 2 & 13; ... ; and 7 & 8.

The number of subsets in each of these categories is just 2 raised to the power of the number of possible elements between the end values, to the total is 2^12 + 2^10 + ... + 2^0.

The binary representation of this is 1010101010101.

The evaluation can be done most easily by recognizing that this is also 1111111 in base 4, which is just 1/3 of 3333333, which in turn is 4^7-1.

So the answer is (4^7 - 1) / 3 = 5461.


  Posted by Charlie on 2010-05-16 12:45:42
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