3
votes
1answer
44 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
1
vote
1answer
42 views

non-negative solutions with upper boundary, in diophantine equation

I wanted to find out in how many ways I can do something, but I don't know combinatorics enough. Can you help me and show or give advice, what we should do in such situations as presented below? Let's ...
0
votes
1answer
31 views

Random nonegative solution of multivariate linear Diophantine equation

Consider a diophantine equation in n variables: $a_1x_1+a_2x_2+...+a_nx_n=k$ All $a_i$'s, $x_i$'s and $k$ are restricted to non-negative integers $\mathbb{Z^+}$. (note that because of domain ...
0
votes
2answers
65 views

number of positive integer solution of inequation

Given an inequation with P,Q,R all integers, $P \cdot R \cdot b + P \cdot Q \cdot c - Q \cdot R \cdot a \geq 0$ how many positive integer solutions of $(a, b, c)$ ? Here $a \leq P, b \leq Q, c \leq ...
1
vote
1answer
56 views

Solution of a simple linear diophantine equation

I'm having a slight problem with a simple equation of the sort $a_1+a_2+a_3...=n$. Where $n,a_1, a_2, a_3... \in N$. I do know how to find the number of solutions to these equations when they are of ...
0
votes
1answer
76 views

How many numbers N satisfy N consecutive positive integers add to 2013?

How would you find how many numbers N there exist such that N consecutive positive integers add to 2013? (Assume that N=1 is a valid case whose solution is just 2013 itself). To clarify, this, when ...
1
vote
2answers
36 views

Combinatorics with 3 equations

Ok, so I want for all combinations of positive integers (a,b,c) for the Equations $$-10(c-2ab)+\frac{b-5}{a}=52$$ $$a-b+c=53$$ $$a(b+7)=54$$ the product abc So I want the product abc not the ...
0
votes
1answer
146 views

Given $p, m$, how many $r, k$ exist such that $\sum_{i=0}^k{m+i \choose p} = {m+r \choose p}$?

I know that ${m+1 \choose p+1} = {m \choose p} + {m \choose p+1}$, does this identity extend further out? My guess is that there exist certain $k$ such that there exists $r > k$ where the title ...
2
votes
2answers
835 views

Number of non-negative integer solutions for linear equations with constants

How do we find the number of non-negative integer solutions for linear equation of the form: $$a \cdot x + b \cdot y = c$$ Where $a, b, c$ are constants and $x,y$ are the variables ?
5
votes
4answers
215 views

Solutions to $a+b+c=12$, $a, b, c \in \mathbb{N}_0$

Let $a, b, c \in \mathbb{N}_0$. If $a+b+c=12$, how many solutions $(a,b,c)$ satisfy the equation? Is the answer:
1
vote
5answers
239 views

Number of solution for $xy +yz + zx = N$

Is there a way to find number of "different" solutions to the equation $xy +yz + zx = N$, given the value of $N$. Note: $x,y,z$ can have only non-negative values.
2
votes
1answer
89 views

Combinatorics of the Zeta function of a variety

I want to know if there is a good combinatorial interpretation of what the Zeta function of a variety $X$ over a finite field $\mathbb{F}_p$ counts. It is defined as $$\exp\sum N_j/j\,t^j,$$ where ...
37
votes
5answers
1k views

Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a certain number of ...
1
vote
2answers
159 views

Integer solutions

How many positive integer solutions are there to $x_1 + x_2 + x_3 + x_4 < 100$? I haven't seen any problems with "less than", so I'm a bit thrown off. I'm not sure if my answer is correct, but ...
0
votes
2answers
2k views

How many integer solutions to a linear combination, with restrictions?

I've already done a few problems such as this, other problems where I'm supposed to find the number of combinations or permutations, subject to certain restrictions. Here's been my basic strategy: ...
-1
votes
1answer
80 views

Count the number of integer solution to $\sum_{i=1}^ {4}{a_i\times b_i} \geq 8 $? [closed]

Count the number of integer solution to $\sum_{i=1}^ {4}{a_i\times b_i} \geq 8 $ such that condition 1: $1 \leq a_i \leq 7$ condition 2: $1 \leq b_i \leq 4$ condition 3: $\sum_{i=1}^{4} {a_i} = ...
0
votes
1answer
109 views

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ [closed]

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ such that condition 1: $1 \leq a_i \leq 7$ condition 2: $1 \leq b_i \leq 4$ condition 3: $\sum_{i=1}^{2} {a_i} = 8$ ...
2
votes
3answers
143 views

What is the number of combinations of the solutions to $a+b+c=7$ in $\mathbb{N}$?

My professor gave me this problem: Find the number of combinations of the integer solutions to the equation $a+b+c=7$ using combinatorics. Thank you. UPDATE Positive solutions
0
votes
1answer
165 views

Count the number of integer solutions for $a \times b \geq k$?

count the number of integer solution for $a \times b \geq k$ given the conditions 1) $1 \leq a \leq p$ 2) $1 \leq b \leq q$ (k, p, and q are constant).
3
votes
2answers
205 views

Count the number of integer solution to $\sum_{i=i}^{n}{f_ig_i} \geq 5 $

How to count the number of integer solutions to $$\sum_{i=i}^{n}{f_ig_i} \geq 5$$ such that $\displaystyle \sum_{i=1}^{n}{f_i}=6$ , $\displaystyle \sum_{i=1}^{n}{g_i}=5$ , $\displaystyle 0 \leq f_i ...
0
votes
1answer
173 views

Total number of solutions of an equation

What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 < x_2 < \cdots < x_r < N$ where $N$ is some natural number and $x_1, ...
2
votes
5answers
133 views

How can I make the following 2 fractions integers?

Let $m,n$ be integers. I want to find the possible values of $m,n$ such that $4(m+n)\over (2m+n)^2+3n^2$ and $4n\over (2m+n)^2+3n^2$ are both integers too. Would someone please help? Of course letting ...
1
vote
3answers
130 views

Number of distinct graphs with y-intercepts that are integers between $-10$ and $10$

I wanted to make a test bank of graphs of linear equations for my algebra classes. I want the $y$-intercept of each graph to be an integer no less than $-10$ and no greater than $10$. Generally, you ...
4
votes
2answers
269 views

How many ordered triple $ (p,a,b) $ is possible such that $p^a=b^4+4$?

If we have a prime number $p$ and two natural numbers $a$ and $b$ such that $p^a=b^4+4$, then how many such ordered triplets $(p,a,b)$ exist? What should be the strategy to solve this one? The only I ...
1
vote
2answers
4k views

How many solutions are there to the equation $x + y + z + w = 17$?

How many solutions are there to the equation $x + y + z + w = 17$? I don't know if I'm doing this right, but I guessed that the solution would be $\binom{20}{3}$, which equals $1140$. Am I doing ...
4
votes
5answers
2k views

Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$

How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$ And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$ In both cases, ...
3
votes
3answers
480 views

Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$

The equation is $(x!)(y!) = x!+y!+z! $ where $x,y,z$ are natural numbers. How to find out them all?
1
vote
1answer
387 views

Number of solutions of Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
5
votes
2answers
514 views

Number of positive integral solutions for $ab + cd = a + b + c + d $ with $1 \le a \le b \le c \le d$

How many positive integral solutions exist for: $ab + cd = a + b + c + d $,where $1 \le a \le b \le c \le d$ ? I need some ideas for how to approach this problem.
2
votes
1answer
3k views

Count the number of positive solutions for a linear diophantine equation

Given a linear Diophantine equation, how can I count the number of positive solutions? More specifically, I am interested in the number of positive solutions for the following linear Diophantine ...
3
votes
2answers
256 views

13 integers with each set of 12 integers

Take 13 integers. Prove that if any 12 of them can be partitioned into two sets of six each with equal sums, then all the integers are the same. Does anyone know if the general case with 2n+1 ...