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| Easy count (Posted on 2014-07-29) |
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Let S be a set of some positive integers.
We'll call S autonomous if the number of elements in S is itself an element of S. e.g. the set {2,3,5} is autonomous, as is the set {2,7}, but the sets {1, 4} or {2,4,5} are not.
Determine a general formula for the number
of autonomous subsets of {1, 2, 3, ... , n}.
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Submitted by Ady TZIDON
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Solution:
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The subset of k numbers must consist of the number k itself + any subset of n-1 members other than k, i.e. one of C(n-1, k-1) possible combinations.
Summing for all possible values of k (1 to n-1):
Sum=C(n-1,0)+ C(n-1,1)+ C(n-1,2)+... C(n-1, n-1)
=2^(n-1).
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