Tagged Questions
2
votes
1answer
28 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 ...
36
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
91 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
293 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:
...
-2
votes
1answer
71 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
90 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
129 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
142 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
154 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
123 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
119 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
113 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
232 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
2k 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 ...
3
votes
5answers
928 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
332 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
298 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
414 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
1k 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
209 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 ...
