Let \( \{x_1, x_2, \ldots , x_n \} \) and \( \{y_1, y_2, \ldots, y_n\} \) be sets of non-negative integers such that \( \sum \limits_{i=1}^n x_i = a \), \( \sum \limits_{i=1}^n y_i = b \), and \(x_i + y_i \geq c\) for all \(i = 1,2,\ldots, n\).
We are also given that . is a positive integer.
Find the number of ordered solutions of the -tuplets of positive integers, .
For example, if , then the number of ordered 6-tuplets of positive integers, is 168.
I wonder if it is possible to count tuples. For example, in this case, because have each, so 2 times is tuples.
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*
or_italics_
**bold**
or__bold__
paragraph 1
paragraph 2
[example link](https://brilliant.org)
> This is a quote
\(
...\)
or\[
...\]
to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
Sort by:
Top NewestThere is no bound for C, that means infinite solutions for the ordered set. It says xi+yi≥c, multiply the whole equation by n and you get nxi+nyi≥c which would mean a+b≥nc, divide the whole equation by n and you have c≤na+b, c could be any number and you can list such positive integers, greater than c .
Log in to reply
I don’t quite get what you want to say. a,b,c are some integers, not all integers. Like the example above, there is a certain number of solutions for some a,b,c.
By the way, thx for coming by :)
Log in to reply
If my doubt is gibberish you can for sure tell me, or maybe you really did not understand what I'm trying to say. Just do share when you find the right solution, or if you already have it.
Log in to reply
I do not have a solution for this. That is why I want to ask this problem.
I would like to know what you mean by 'you can list such positive integers, greater than c .'
By the way, have you tried out the example? That might give some sense what this problem is about
Log in to reply