# Questions tagged [multisets]

For questions about or related to multisets, a notion similar to sets with the difference that elements can be repeated.

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### Is it possible to show that no multiset may exist to satisfy a given constraint without knowing the multiplicity of the set's elements?

I'd like to preface my question by stating that I have no education in mathematics beyond A levels (UK), which was 7 years ago, so I apologise if I'm asking a silly question or if I misuse any terms. ...
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### Let $n$ be a positive integer. Determine the number of solutions to $x_1 + \cdots + x_k \leq n$ with nonnegative integer solutions.

Let $n$ be a positive integer. Determine the number of solutions to $x_1 + \cdots + x_k \leq n$ with nonnegative integer solutions. Determine the number of solutions with positive integer solutions. I ...
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### Analysing ways to choose $m$ naturals from set of first $n$, s.t. they differ by $\ge k$.

Ways to choose $m$ distinct natural numbers from set of first $n$ natural numbers, s.t. the chosen natural numbers differ by at least $k$. Request vetting for my approach: The numbers to be chosen are ...
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### Counting the number of distinct groups with and without repeat items

Here's a simple question that popped into my head. I shouldn't be struggling with but am. Suppose there are $n$ objects. We want to find the number of distinct groups. First let's take the case where ...
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### Expected number of distinct elements when drawing from a multiset with replacement

Suppose I have a multiset $M$ with elements $1,\dots,m$ and respective multiplicities $n_1, \dots, n_m$: $$(1, n_1) \\ (2, n_2) \\ \dots \\ (m, n_m)$$ What is the expected number of distinct ...
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### Number of combination with repetitions containing a subset of all the root element

I am in trouble with this question: Given the set $S$={$a,b,c,d,e,f$} and 10 position to arrange these elements. How many combination with repetitions exist if the only allowed must contain at least ...
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### Finding the R-minimal and R-maximal

For any two integers $a,b$, we say that $a$ divides $b$ (written as $a\mid b$) if $b=ak$ for some $k∈\Bbb Z$. Let $A=\{3,6,7,9,12,14,21,42,252\}$. Consider the partial order relation $R$ on $A$ given ...
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### Determining what relation means

If we have to consider the relation R from {1,2,3,4} to {3,4,6,7,9} given by aRb ↔ b = a + 3 then would the relation given be {1,2,3,4,6,7,9}? I'm a little confused on what is meant when I'm being ...
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### Counting permutations of multisets obtained from the prefix sum of certain integer arrangements

Consider distinct arrangements of $k$ nonnegative integers that sum to $s$ with the the additional condition that the sum of every other integer is $t \le s$. A bijective mapping of these arrangements ...
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### Finding relation of two sets

If A = {1, 2, 3} and B = {1, 2, 3, 4}, would R = {(a, b) ∈ A × B | b = a^2} be {(1,1), (2,4)} since the only time that b = a^2 is true is when (A,B) = (1,1) and (A,B) = (2,4)?
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### How can i resolve this point (a,b) combinations?

In a given programming language, an identifier is a sequence of a certain number of characters in which the first character must be a letter of the English alphabet and the rest can be a letter or a ...
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### In how many ways can 15 identical math books be distributed to six students?

question: In how many ways can 15 identical math books be distributed to six students? i try do this: P(15,6) could be this the answer or i should check the question, i find out other? help please
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### Defining the Set of Rational Numbers

I was wondering about how to define the set of rational numbers, as I am currently learning about set theory in a class of mine. We were going through using set builders to define sets and produced ...
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### How many groups of pentagonal flower bouquets can be formed?

A florist has three types of flowers: tulips, roses, and daisies. There are 4 tulips, 5 roses, and 6 daisies. These 15 flowers are to be arranged into three bouquets of 5 flowers each. Assume that ...
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### Some sum of two elements exceeds a third one in a finite multiset with bounded values

For a multiset $S = \{ a_1,a_2,a_3,\ldots,a_k \}$ where $k = 13$ and $1 \leqslant a < 32$, prove that there exists a subset $s = \{ a_i, a_j, a_k \}$ such that each sum of two elements exceeds the ...
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### How do you translate 'positions of a value 2' into a mathematical notation when defining a bijection?

Let's say I have a multi-set: $(1,1,2)$ And I want to define a bijection from $A$ to $B$ such that: $(1,1,2)$ becomes {3} and, $(1,1,1)$ becomes {} and, $(2,2,2)$ becomes {1,2,3} or, in general, it is ...
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### Find number of ways to divide a set into 2 parts

In how many ways can we divide a set into 2 parts having an element in equal number in both of resulting subsets. For example, multiset = {1, 2, 3, 5, 5, 5, 5} and ...
Let $a,b,c,d,e,f,g,h$ be natural numbers such that the multisets $\{a,b,c,d,a+b,c+d\}$ and $\{e,f,g,h,e+f,g+h\}$ are the same. Can we say that $\{a,b\}=\{e,f\}$ or $\{g,h\}$ ? and similarly $\{c,d\}=\{... 1answer 237 views ### Permutation with Repetition Index Conversion I'm looking for the equation to determine the index of a permutation with repetition with known parameters. For example: A total of$9$values,$4$A's and$5$B's Gives a total of$126\$ permutations ...
Let's say I have a multiset of integers a with a size n (here n = 10) $$a = \{1, 1, 2, 2, 3, 4, 5, 6, 6, 10\}$$ I'd like to know ...