Linked Questions
95 questions linked to/from Extended stars-and-bars problem(where the upper limit of the variable is bounded)
5
votes
1
answer
1k
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Number of combinations of two numbers from a list with repeating numbers? [duplicate]
I've tried googling it and looking it up on this website but since I don't know the technical term for this calculation I ran out of luck. Basically, if I have a collection of numbers (each of which ...
- 350
2
votes
2
answers
470
views
Non-negative integers to form a sum with restrictions? [duplicate]
How do I solve this?
Number of non-negative solutions to $x_1 + x_2 + x_3 + x_4 = 4$ where $0 \le x_i \le 3$?
What's the general technique? I already know the technique for $j \le x_i$ but have no ...
- 21
1
vote
2
answers
354
views
Possible ways to have $n$ bounded natural numbers with a fixed sum [duplicate]
Is it possible to count in an easy way the solutions of the equations and inequalities $x_1+x_2+\cdots+x_n = S$ and $x_i\leq c_i$ if all $x_i$ and $c_i$ are natural numbers?
- 4,852
2
votes
1
answer
350
views
Count Subsets of size less than equal to k [duplicate]
This is a variation of question asked on this site before.
Consider a set with $π_1$ 'distinct' 1s, $π_2$ 'distinct' 2s, ... , $π_π$ 'distinct' ns. You have $π_1+1$ choices for the 1s (including ...
user680738
3
votes
1
answer
216
views
Number of solutions for an equation with constraints on each variable in the equation [duplicate]
I have to find the number of solutions for: $$x_1 + x_2 + x_3 + x_4 = 42$$
when given:
$$ (I) 12 <= x_1 <=13 $$
$$ (II) 3 <= x_2 <= 6 $$
$$ (III) 11 <= x_3 <= 18 $$
$$ (IV) 6 <= ...
- 4,861
1
vote
2
answers
153
views
Distributing $60$ identical balls into $4$ boxes if each box gets at least $4$ balls, but no box gets $20$ or more balls [duplicate]
How many different ways can the balls be placed if each box gets at least $4$ balls each, but no box gets $20$ or more balls?
I was thinking about finding all the possible ways which every box gets ...
- 11
0
votes
2
answers
63
views
Elementary Combinatorics Questions [duplicate]
I'm trying to understand a psychology study I've just read and I'm encountering a combinatorics problem that I can't solve and could use some help with.
There is a questionnaire that asks 5 questions,...
- 1,836
0
votes
1
answer
124
views
Number of non negative integer solutions of $l_1+l_2+...+l_m=n$ for constraint $l_i\leq k$ [duplicate]
How do I find a closed form solution to number of non negative integer solutions of $l_1+l_2+...+l_m=n$ for constraint $l_i\leq k$. I know that without the constraint, the answer is $$n+m-1 \choose{m-...
- 2,457
0
votes
1
answer
83
views
Combinatorics: How to choose k objects from limited number of choices? [duplicate]
Suppose we want to choose a certain number of objects, k, from a variety of choices, all limited. I'm confused about how to set up this problem. I'm pretty sure it's stars and bars, but I don't know ...
- 1
1
vote
1
answer
65
views
How to calculate compositions when the numbers cannot be greater than a certain value? [duplicate]
The number of $k$-compositions of a positive integer $n$ is ${n-1}\choose{k-1}$, and its number of $k$-weak compositions is ${n+k-1}\choose{k-1}$.
However, how to calculate the corresponding ...
- 865
0
votes
1
answer
63
views
Formula for calculating the combination of a multiset taken r at a time? [duplicate]
If we have a multiset S = {a,a,b,b,b,c,d}
How to calculate all possible combinations if we take r items at a time?
For example if r = 3 then the combinations will ...
0
votes
0
answers
60
views
Adding weak $d$-compositions of $n$ [duplicate]
Let $\vec k=(k_1,...,k_d)$ be a $d$-tuple with $k_i$ being non-negative integers satisfying $\sum_i k_i=n$; $\vec k$ is then referred to as a weak $d$-composition of $n$. It is clear that if $\vec a$ ...
- 1,835
0
votes
1
answer
60
views
k numbers adding up to n [duplicate]
Suppose I have following equation:
$$ a_1 + a_2+\cdots+a_k=n$$
$$a_1,a_2,\cdots,a_k\in\{m,m+1,m+2,\cdots,m+l \}$$
and
$$m*k<n$$
I am asked to calculate how many different solution that equation ...
- 457
1
vote
0
answers
60
views
Counting the number of integer solutions to a simple equation [duplicate]
In the following equation with unknown integers $x_i$, $1 \leq i \leq N$, the sum of all those integers are $R$. A constraint is added to each integer such that $Min_i \leq x_i \leq Max_i$.
The ...
- 11
4
votes
0
answers
45
views
Number of unique ways to select k objects from n objects, when some of the objects are identical. [duplicate]
So, the number of ways to select $k$ objects from $n$ unique objects is equal to the binomial coefficient $\binom{n}{k}$.
Now assume that not all $n$ objects are unique, but there are $m$ groups of ...
- 143