Question

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.

$256$

$136$

$108$

$144$

$156$

Answer: D

Concepts:combinations casework

Difficulty rating: 1570

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 $\begin{aligned} \sum_{m\ge 1}\binom{12-m}{m-1} &=\binom{11}{0} \\ &\quad {}+\binom{10}{1} \\ &\quad {}+\binom{9}{2} \\ &\quad {}+\binom{8}{3} \\ &\quad {}+\binom{7}{4} \\ &\quad {}+\binom{6}{5}, \end{aligned}$ which equals $1+10+36+56+35+6$ $=144.$

Thus, the correct answer is D.

2023 AMC 12B Problem 8

Answer: D

Difficulty rating: 1570

Solution:

Problem 8 in Other Years