# Tagged Questions

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### Generating Functions and Linear Diophantine Inequalities

The following exercise is from Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick, page 46. A $k$-composition of $n$ is an ordered $k$-tuple of non-negative integers whose sum is $n$. ...
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### 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?
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### 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 ...
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### 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 ...
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### 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: $\sum_{i=1}^{2} {a_i} = 8$ ...
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### 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
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### 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).
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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?
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### 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 ...
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### 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.