2023 AMC 12B Problem 8

Below is the professionally curated solution for Problem 8 of the 2023 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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Concepts:combinations casework

Difficulty rating: 1570

How many nonempty subsets $B$ of $\{0,1,2,3,\ldots,12\}$ 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.

$256$

$136$

$108$

$144$

$156$

Solution:

If the least element is $m\ (m\ge 1),$ then $|B|=m$ and the remaining $m-1$ elements come from $\{m+1,\ldots,12\},$ a set of size $12-m.$ The count is $\sum_{m\ge 1}\binom{12-m}{m-1}=\binom{11}{0}+\binom{10}{1}+\binom{9}{2}+\binom{8}{3}+\binom{7}{4}+\binom{6}{5},$ which equals $1+10+36+56+35+6=144.$

Thus, the correct answer is D.

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