2023 AMC 12B Problems/Problem 8
Problem
How many nonempty subsets of have the property that the number of elements in is equal to the least element of ? For example, satisfies the condition.
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 . 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 . 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 . This is .
~lprado
Note:
In general, i.e. when the number is not , the answer is . -Mr Sharkman
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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