# Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

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### Answer to a permutation and combination question.

The question is 'How many different whole numbers are factors of number 2x3x5x7x11x13 ?' My answer to this question is 63 but the right answer is 64. I don't know why it is 64? I need some assistance.
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### How many permutations are there for PARALLELOGRAM where the A's are separated?

In my book there is a permutation example question, the question is: How many permutations are there of the letters of the word PARALLELOGRAM? In how many of these are the A's separated? I ...
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### Selecting exactly $2$ men given at least one woman is selected.

A committee of four is to be chosen from a group of four men and five women. What is the probability that the committee contains exactly two men given that at least one woman is chosen? I have tried ...
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### Determining Probability of Multiple Dies with Similar Faces

I have the case where I know what to do with normal equally fair dice, especially in succession, but I have a situation where I have many dice that have varying numbers of similar faces, and I'm ...
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### Combinations, a Deck with Separate Hands and Players

The main issue I'm having is trying to solve this scenario below where we have multiple decks between multiple players, with left over cards that remain in no player's hand. With this I've tried to ...
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### How to use exponential generating functions to count the number of k-letter permutations from n letters?

I was learning about exponential generating functions and stumbled upon the following question and answer : Sample question Given the string of letters ABBBBBBBBBBBBBBBBBCDEFGHIJKLMOPQRSTUVWXYZ (that'...
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### Combinatorics, 2-Tree, Sequence

I've just thought about a combinatoric problem. Say you have a tree with $n$ nodes at the $n$-th level ($2$-tree). Number elements based on their position left to right, top to bottom. Let $a_{n,i}$ ...
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### Calculating a maximum size of subset of permutations and finding an example of such subset. [closed]

I've been trying to solve a problem of finding a subset of permutations under a certain constraint. So far I wasn't able to solve this, hope someone can help. Thank you in advance. The problem: We ...
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### How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child?

Problem How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child? The seats are not numbered. My Opinion I ...
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### I am trying to generate random matrix with based on some condition, How many matrices the can be generated?

I am trying to generate random matrices based on the following conditions. There will be 3X3 matrices. The first column can have 1 to 10, the second column can have 11 to 20 and the third column can ...
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### Find the number of ways of tiling a $3\times n$ rectangular grid with $2\times 1$ dominoes

I'm trying to find the number of ways $(a_n)$ of tiling a $3\times n$ rectangular grid with $2\times 1$ dominoes, where rotation is allowed. I want to find a recurrence relation for $(a_n)$ and an ...
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### Find the number of elements in $\{0,1\}^n$ with no more than three $1$'s or three $0$'s in a row

I'm trying to find a general formula for the number of elements $s_n$ in $\{0,1\}^n$ with no more than three $1$'s or three $0$'s in a row, where $n\geq1$. I calculated $s_n$ for small values of $n$ ...
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### How many combinations does this cube have and what is the probability that we get all of the same colour?

I recently came across a contraption. For those who can’t access the picture I have attached, I will include the following description. There is a concealed contraption the shape of a large cube. You ...
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### Probability of pair of gloves selection

In his wardrobe, Fred has a total of ten pairs of gloves. He had to pack his suitcase before a business meeting, and he chooses eight gloves without looking at them. We assume that any set of eight ...
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### SET Card Game | Probability that no SET can be found in $12$ randomly chosen cards. [duplicate]

PROBLEM The question is related to SET Card Game. I was thinking about that if I choose $12$ cards randomly from a deck then what will be the probability of no $3$-card-SET being there. MY METHOD It ...
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### Prove that there is a circle containing exactly $2018$ points

Problem Given a set $\mathtt{E}$ containing $2017^{2019}$ points on the plane. Prove that there is a circle containing exactly $2018$ points from the set $\mathtt{E}$ (these points are on the open ...
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### Binomial PDF where I have my $n,$k and RHS result – but need to reverse engineer the $p$ and (by extension) $1−p$.
I do not know how to do this soundly. I can arrive at the answer if I iterate the formula with $p$ in a range from $0$ to $0.99$ but I want to know a mathematical way to arrive at it. The R code for ...
I was going through a paper where I stuck on a combinatorial argument as follows I want help with the first assertion i.e proving the inequality $\alpha(m+l)\le\alpha(m)$. As the author suggests it is ...