How many mn tuples are there??

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 ca+bnc \leq \frac{a+b}n . cc is a positive integer.

Find the number of ordered solutions of the 2n2n-tuplets of positive integers, (x1,x2,,xn,y1,y2,,yn)( x_1, x_2, \ldots, x_n, y_1, y_2, \ldots , y_n ) .

For example, if n=3,a=4,b=6,c=2n=3,a=4,b=6,c=2, then the number of ordered 6-tuplets of positive integers, (x1,x2,x3,y1,y2,y3)(x_1, x_2, x_3, y_1, y_2, y_3) is 168.

I wonder if it is possible to count mnm\cdot n tuples. For example, in this case, m=2m=2 because xi,yix_i, y_i have nn each, so 2 (=m)(=m) times nn is 2n2n tuples.

Note by Inquisitor Math
3 months, 2 weeks ago

No vote yet
1 vote

  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:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

There is no bound for C, that means infinite solutions for the ordered set. It says xi+yicx_i+y_i≥c, multiply the whole equation by nn and you get nxi+nyicnx_i+ny_i≥c which would mean a+bnca+b≥nc, divide the whole equation by nn and you have ca+bnc≤ \dfrac{a+b}{n}, 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 :)

Inquisitor Math - 3 months ago

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

Inquisitor Math - 3 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...