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Difference between revisions of "2023 AMC 12B Problems/Problem 8"

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==Problem==
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How many nonempty subsets B of <math>{0, 1, 2, 3, \cdots, 12}</math> have the property that the number of elements in B is equal to the least element of B? For example, B = <math>{4, 6, 8, 11}</math> satisfies the condition.
  
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<math>\textbf{(A) } 256 \qquad\textbf{(B) } 136 \qquad\textbf{(C) } 108 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 156</math>
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==Solution 1==
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There is no way to have a set with 0. If a set is to have its lowest element as 1, it must have only 1 element: 1. If a set is to have its lowest element as 2, it must have 2, and the other element will be chosen from the natural numbers between 3 and 12, inclusive. To calculate this, we do <math>\binom{10}{1}</math>. If the set is the have its lowest element as 3, the other 2 elements will be chosen from the natural numbers between 4 and 12, inclusive. To calculate this, we do <math>\binom{9}{2}</math>. We can see a pattern emerge, where the top is decreasing by 1 and the bottom is increasing by 1. In other words, we have to add <math>1 + \binom{10}{1} + \binom{9}{2} + \binom{8}{3} + \binom{7}{4} + \binom{6}{5}</math>. This is <math>1+10+36+56+35+6 = \boxed{144}</math>.
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~lprado

Revision as of 18:16, 15 November 2023

Problem

How many nonempty subsets B of ${0, 1, 2, 3, \cdots, 12}$ have the property that the number of elements in B is equal to the least element of B? For example, B = ${4, 6, 8, 11}$ satisfies the condition.

$\textbf{(A) } 256 \qquad\textbf{(B) } 136 \qquad\textbf{(C) } 108 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 156$

Solution 1

There is no way to have a set with 0. If a set is to have its lowest element as 1, it must have only 1 element: 1. If a set is to have its lowest element as 2, it must have 2, and the other element will be chosen from the natural numbers between 3 and 12, inclusive. To calculate this, we do $\binom{10}{1}$. If the set is the have its lowest element as 3, the other 2 elements will be chosen from the natural numbers between 4 and 12, inclusive. To calculate this, we do $\binom{9}{2}$. We can see a pattern emerge, where the top is decreasing by 1 and the bottom is increasing by 1. In other words, we have to add $1 + \binom{10}{1} + \binom{9}{2} + \binom{8}{3} + \binom{7}{4} + \binom{6}{5}$. This is $1+10+36+56+35+6 = \boxed{144}$.

~lprado

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