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Cardinality is the Least Element

Source: 2023 AMC 12B #8

November 15, 2023
AMCAMC 122023 AMC2023 AMC 12B

Problem Statement

How many nonempty subsets BB of {0,1,2,3,,12}\{0, 1, 2, 3, \dots, 12\} have the property that the number of elements in BB is equal to the least element of BB? For example, B={4,6,8,11}B = \{4, 6, 8, 11\} satisfies the condition.
<spanclass=latexbold>(A)</span> 256<spanclass=latexbold>(B)</span> 136<spanclass=latexbold>(C)</span> 108<spanclass=latexbold>(D)</span> 144<spanclass=latexbold>(E)</span> 156<span class='latex-bold'>(A)</span>\ 256 \qquad<span class='latex-bold'>(B)</span>\ 136 \qquad<span class='latex-bold'>(C)</span>\ 108 \qquad<span class='latex-bold'>(D)</span>\ 144 \qquad<span class='latex-bold'>(E)</span>\ 156