![Questions Tagged [Combinatorics]](https://docuprint.cloud68.co/uploads/questions-tagged-combinatorics_1200.jpg)
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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Pigeon hole tennis problem
While you are on a 4-week vacation, you will play at least one set of tennis matches each day, but you wouldn’t play more than forty sets total during this time. Prove that no matter how you ...
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Probability of markov chain in a finite set
Let's $X$ be an homogeneous Markov chain with three states $\{1,2,3\}$ et denote $(\pi_1,\,\pi_2,\,\pi_3)$ the initial probabilities and $P=(p_{ij})_{1\leq i,j\leq3}$ the transition matrix.
Let's ...
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Exponential generating function of the relation $B(n,k)=B(n-1,k-1)+(n-1)B(n-2,k-1)$ —just over $n$—
Let $B(n,k)$ the number of permutations of the set $[n]=\{1,\ldots,n\}$ that are decomposable in $k$ disjoint cycles of order $1$ or $2$. For example, $\mu=(1,3)(2,5)(4)(6,7)(8)$ is counted by $B(8,5)$...
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In how many ways can the committee be selected if the committee must have at least one member of each sex? [duplicate]
A committee of $5$ is to be formed from $5$ men and $5$ women. In how many ways can the committee be selected if the committee must have at least one member of each sex?
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Probability edge in matching contain element from two disjoints subsets $\mathbb{P}(\exists \{i,j\} \in M, i \in S_1 \text{ and } j \in S_2) \geq 1/2$
Let $n$ be an even perfect square. First generate a random sample $S$ without replacement choosing $2 \sqrt{n}$ numbers from $[n]$.
$$S = \{s_1,...,s_{2\sqrt{n}}\}.$$
Then split $S$ in half. So we get ...
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How many five-character sequences composed of lower case letters and digits be formed if letters may be repeated but digits cannot be repeated?
Permutation/Combination where some elements, such as lower case character of the alphabet a-z are repeating, and some are non-repeating such as number of 0-9, for a string that is 5 characters long.
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Find $X/1430$ when $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$
Let $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$, then what's the value of $X\over1430$?
I don't even know where to begin on this question. All solutions I've seen on various ...
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Partitioning vertices and deriving an upper bound for the number of edges of a vertex subset
Let $G=(V,E)$ be a graph on $2n$ vertices i.e. $|V(G)|=2n$. Moreover, the graph $G$ is assumed to be a $3$-partite graph such that $V(G)=\bigcup_{i=1}^{3}V_i$ where $V_i$ denotes the $i$th partite set....
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Choose cards in combination [duplicate]
A decks of cards consisting of 8 red ,8 yellow ,8blue,8 green cards.Cards of the same colours are numbered 1,2.....8.How many ways can 6 cards be chosen such that at least one card of each colour is ...
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Chasing a monster on a $3 \times 3$ grid
There are nine rooms as shown below $$\begin{array}{|c|c|c|}\hline1&2&3\\\hline4&5&6\\\hline7&8&9\\\hline\end{array}.$$ A monster is in one of the rooms. Each turn, the people ...
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Is the following combinatorial relation correct?
I am confused regarding the following problem in combinatorics ( statistical mechanics ).
Suppose I have the following relation : $$\sum_{i=1}^N n_i=\bar{N}$$
I have to find out the number of possible ...
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Calculate Cov(X,Y) where X is #children with no books and Y is #children with exactly 1 book while distributing r books to n children.
Suppose $r\geq 1$ distinct books are distributed at random among $n\geq 3 $ children. Let $X$ be the number of children who do not get any book, and $Y$ be the number of children who get exactly one ...
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A tight upper bound on this Binomial sum
I have the following function:
$P(n)=q^n\sum\limits_{H=0}^{n-1}{{H+n-1\choose H}w^H}+w^n\sum\limits_{H=n}^{\infty}{{H+n-1\choose H}q^H}$, where $0<q<0.5<w<1$ and $q+w=1$.
My end goal is to ...
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Finding a bound on a certain number of sums
Let $x_1, ..., x_{2n}$ be real numbers with $|x_i
| ≥ 1$ for all $i$, and let $I ⊂ R$ be an
arbitrary open interval of length $2$. I want to:
(a) Prove that the number of sums $\sum_{i=1}^{2n}ε_ix_i$
,...
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How to find the number of options for choosing numbers from $a_1, a_2, a_3, ... a_n$ such that their sum was equal to $k$
Let our numbers $2, 5, 6, 7, 10, 15$ and $k = 15$. I need to find the number of possible options for choosing numbers that form a total of 15. It's $(5, 10), (2, 7 ,6), (15)$. So the answer is 3.
