Linked Questions
52 questions linked to/from Number of solutions to equation of varying size, with varying upper-bound range restrictions
33
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5
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Counting bounded integer solutions to $\sum_ia_ix_i\leqq n$
I want to find the number of nonnegative integer solutions to
$$x_1+x_2+x_3+x_4=22$$
which is also the number of combinations with replacement of $22$ items in $4$ types.
How do I apply stars and bars ...
0
votes
0
answers
70
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Find The Number Of Solutions To The Following Equation [duplicate]
Find the number of integer solutions to the equation:
$x(1) + x(2) + ··· + x(11) = 49 $
such that $0 \leqslant x(r) \leqslant 8$ for each $r= 1, 2,...,8$
I Used Principle Of Inclusion And Exclusion To ...
0
votes
0
answers
54
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Stars and Bars with constraints on the variables. [duplicate]
How do I find the number of solutions in the equation $$a_1 + a_2 + a_3 +\dots+a_k=2021$$ in terms of $k$ with $1\leq a_1, a_2, \dots, a_k\leq9$? I know the first step is to introduce a new variable, ...
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0
votes
0
answers
28
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Drawing 6 chocolates from 15 chocolates with given constraints [duplicate]
Number of ways we can draw 6 chocolates drawn from 15 chocolates out of which 4 are blue, 5 are red and 6 are green if chocolates of the same colour are not distinguishable?
what i did was to use the ...
- 977
30
votes
3
answers
12k
views
Extended stars-and-bars problem(where the upper limit of the variable is bounded)
The problem of counting the solutions $(a_1,a_2,\ldots,a_n)$ with integer $a_i\geq0$ for $i\in\{1,2,\ldots,n\}$ such that
$$a_1+a_2+a_3+\ldots+a_n=N$$
can be solved with a stars-and-bars argument.
...
1
vote
2
answers
803
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Inclusion–exclusion principle; what is $(-1)^{n+1}$
could somebody kindly confirm that my understanding of inclusion-exclusion matches it's formula.
for a 3 sets example; we add 3 unions, subtract the total of all 3 pairwise intersections and add the ...
- 261
1
vote
1
answer
1k
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How do I find the generating function for the sequence $-2, 4, -8, 16, -32, 64, ...$
I am trying to find the generating function for the sequence: $-2, 4, -8, 16, -32, 64, ...$ and I am not sure what the answer is.
I have colleagues who are saying that $g(x) = \frac{1}{(1+2x)}$ is the ...
- 417
3
votes
1
answer
595
views
How many 7 digit numbers have a digit sum of 11
I had this challenge question for my perms and combs homework, but I am a bit unsure on how to go about solving it.
How many $7$-digit numbers have a digit sum of $11$?
To do this problem, I tried ...
- 322
3
votes
3
answers
208
views
Partial Fraction of $\frac{1-x^{11}}{(1-x)^4} $ for Generating Function
The original question involves using generating functions to solve for the number of integer solutions to the equation $c_1+c_2+c_3+c_4 = 20$ when $-3 \leq c_1, -3 \leq c_2, -5 \leq c_3 \leq 5, 0 \leq ...
- 41
4
votes
2
answers
394
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Generating series for ternary strings without 000 and not ending with 0
I would like to find a formula for $T_n$, the number of ternary strings of length $n$ so that they do not contain three consecutive zeroes, and they do not end with $0$ as well. I can find a ...
- 685
1
vote
1
answer
599
views
Probability of Bingo on a 3x3 grid
Consider a $3\times 3$ grid
We paint 3 grids yellow, 3 grids green and 3 grids red
What is the probability of that there exists three consecutive grids that are the same color (aka "bingo"), ...
- 839
2
votes
1
answer
394
views
Is there a formula that gives the number of compositions of length $k$ of a number $n$ containing numbers no higher than $p$?
A composition of a positive integer $n$ is a way of writing $n$ as the sum of an ordered sequence of strictly positive integers. Any positive integer $n$ has $2^{n - 1}$ distinct compositions.
...
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1
vote
2
answers
349
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How many ways can we write $N$ as a sum of $K$ strictly positive numbers?
I know this has been asked before, but most existing answers have been in the form of summations instead of framing this as a stars and bars problem as taught in class.
We were given that the answer ...
- 21
4
votes
2
answers
257
views
Number of solutions to equation, range restrictions per variable
Find the number of solutions of the equation $x_1+x_2+x_3+x_4=15$ where variables are constrained as follows:
(a) Each $x_i \geq 2.$
(b) $1 \leq x_1 \leq 3$ , $0 \leq x_2$ , $3 \leq x_3 \leq 5$, $2 \...
- 191
5
votes
1
answer
183
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Distributing $10$ indistinguishable cars, $12$ indistinguishable balls, $14$ indistinguishable teddy bears to $3$ children, each have exactly $7$ toys
I have $10$ indistinguishable cars , $12$ indistinguishable balls, $14$ indistinguishable teddy bears.I want to distribute them to $3$ different children in a kindergarten such that each child will ...
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