Difference between revisions of "2023 AMC 12B Problems/Problem 8"
Failure.net (talk | contribs) (→Solution 1) |
Mr.sharkman (talk | contribs) |
||
Line 8: | Line 8: | ||
~lprado | ~lprado | ||
− | + | ==Note:== | |
+ | In general, i.e. when the number is not <math>13</math>, the answer is <math>F_{n-1}</math>. | ||
+ | -Mr Sharkman | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2023|ab=B|num-b=7|num-a=9}} | {{AMC12 box|year=2023|ab=B|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 16:07, 16 November 2023
Contents
[hide]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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.