Question 1210202: How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step by step.
**1. Understand the Constraints**
* We are looking for subsets of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
* Each subset must have at least two elements.
* The sum of the two largest elements in each subset must be 4.
**2. Identify the Possible Pairs**
Since the sum of the two largest elements must be 4, the possible pairs of largest elements are:
* (0, 4)
* (1, 3)
* (2, 2)
**3. Analyze Each Pair**
* **Pair (0, 4):**
* The subset must contain 0 and 4.
* Other elements can be any subset of {}.
* There is only one subset: {0, 4}.
* **Pair (1, 3):**
* The subset must contain 1 and 3.
* Other elements can be any subset of {0}.
* Possible subsets: {1, 3}, {0, 1, 3}.
* There are 2 subsets.
* **Pair (2, 2):**
* The subset must contain 2.
* Other elements can be any subset of {0, 1}.
* Possible subsets: {2, 2} which is just {2}, {0, 2}, {1, 2}, {0, 1, 2}.
* However, we need at least two elements, so we must exclude {2}.
* Possible subsets: {0, 2}, {1, 2}, {0, 1, 2}.
* There are 3 subsets.
**4. Count the Subsets**
* For (0, 4): 1 subset
* For (1, 3): 2 subsets
* For (2, 2): 3 subsets
Total number of subsets = 1 + 2 + 3 = 6
**Therefore, there are 6 subsets of {0, 1, ..., 9} that have the property that there are at least two elements and the sum of the two largest elements is 4.**
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?
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Mathematically, the problem's formulation is incorrect, ambiguous and self-controversial.
In other words, this given formulation in the post is mathematically illiterate.
In addition, from the solution by @CPhill, it is clear, that the author does not understand
and incorrectly uses/treats the notion/conception of a subset.
He mistakenly thinks, for example, that {1,3} and {3,1} are different subsets and counts them
as different subsets, while it is THE SAME subset, which should be counted only once.
Also, the subset of three elements {0,1,3}, is missed and is not counted.
So, the post by @CPhill is not a solution.
It is an example on how this problem should not be solved.
So, in the 5-score scale, I give the score '2' for the problem's formulation, and the score '3' for the solution,
where '3' is for enthusiasm, not for the correctness of the solution.
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Hello, I perfectly understand that all your efforts (from your side) are targeted to fill the database for your AI.
I want to say, that the person/the persons who are responsible for problems formulations on your side,
have no an adequate mathematical level and should be REPLACED.
Regarding Edwin's stories about the chess AI, I must to say that the chess AI
was developed by the teams of highest level SPECIALISTS and PROFESSIONALS
in both chess and programming and with fundamental Math background.
Their level was incomparably higher than what I have been observing here
at this forum day after day just for about 14 years.
To compare their levels is the same as to have a 3-score student in Math, from one side,
and a mathematician, laureate of Fields medal, from the other side.
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Now I will explain you, what efforts are needed in order for a person has a professional, clear and proper Math language.
The base is "all 5 in Math" in the middle and high school
with the full knowledge of the material from good teachers.
It is desired/necessary to have a special Math education from Math school during these years.
It is desired/necessary to participate in Math circle during these years.
Or to read several (5 - 10) popular Math books above a standard curriculum
and solve ~ 1000 problems from 5 books-collections of Math problems above a standard school Math curriculum.
Participate at Math Olympiads at the level at least of local University or higher.
If the number of participants is about 1000-2000 of your age,
it is desired you be among top 20 - 50 winners.
Then 4 or 5 years studying Math as a major at a University/college.
After that: ~4 years working on PhD in Math or Science, reading tens and hundreds articles in Math
and reading tens of relevant textbooks and monographs.
Then EITHER
After getting the PhD degree - 5 or 10 years working in Math or Science, writing reports and
peer-reviewed articles and making professional presentations and expositions.
OR
Teaching Math for 10 years and preparing/publishing textbooks (or participating in preparing/publishing textbooks).
When a person has this background behind his or her shoulders, he or she can work
as a leader (or as one of the leaders in the team) for making AI in the area of Math education.
So, it requires not a year, not two years, not three years and even not 5 years preparation.
It requires at least (or about) 20 years from a school to the leader's position in such a team.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The AI solution from the other tutor is wrong; a subset of the integers from 0 to 9 inclusive can't contain the same element twice.
The requirements are that the subset must contain at least two elements and the sum of the two largest elements is 4.
(1) If the largest element in the subset is 4 and the sum of the two largest elements is 4, then the second largest element in the subset is 0. Then, since 0 and 4 are the two largest elements, there can be no other elements in the subset.
So the subset {4,0} satisfies the conditions.
(2) If the largest element in the subset is 3 and the sum of the two largest elements is 4, then the second largest element in the subset is 1. Then, since 3 and 1 are the two largest elements, the subset can either contain or not contain the element 0.
So the subsets {3,1} and {3,1,0} satisfy the conditions.
There are no other subsets that satisfy the conditions.
ANSWER: There are three subsets that satisfy the conditions:
{4,0}
{3,1}
{3,1,0}
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