# Tagged Questions

For questions about the study of finite or countable discrete structures, specifically 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.

90 views

54 views

### How many ways are there to add up odd integers to 20?

How many ways are there to add up odd integers to 20? Here, $1+19$ is one solution, $19+1$ is a different solution, and $1+1+\dots+1$ counts as just one solution.
57 views

### Incompressible countable Total-ordering implies well-ordering i

A totally-ordered set (S,<) is incompressible if $(S,<) \cong (T,<)$ and $S \supseteq T$ implies $S = T$. Is it true that if $S$ is incompressible countable and totally-ordered then $S$ is ...
63 views

### Number Theory and p-Remainder Numbers

In order to submit the problem, here it comes the definition we are interested in. Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$ and some natural $p > 1$, we will designate a p-remainder ...
38 views

### Number of ways to arrange $8$ rooks on a chessboard

Find the number of ways to arrange $8$ rooks on a chessboard such that no two of them attack other? I was thinking it would be $64 \times 49 \times 36 \times 25 \times 16 \times 9\times 4 \times 1$, ...
45 views

### Proof verification: Mantel's theorem

if a graph $G=(V,E)$ on $n$ vertices contains no triangles than it contains at most $n^2/4$ edges. Proof: Let v$\in$V be a vertex of maximum degree k. since G contains no triangles, there are no ...
43 views

### Different ways to create a 5 different digit integer

The number of ways to create a five digit integer such that all of the digits are different is $9 \cdot 9 \cdot 8 \cdot 7 \cdot 6 = 27216$. However, if I select the numbers from the ones digits first, ...
23 views

### Covering a uniform hypergraph with complete $r$-partite hypergraphs

In combinatorial terms, I was wondering how many complete $r$-partite $k$-uniform hypergraphs are needed to cover the edges of the complete $n$-vertex $k$-uniform hypergraph $\binom{[n]}{k}$. An $r$-...
95 views

379 views

### Arranging numbers around a square

In how many ways numbers 1 to 12 can be arranged on a sides of squares (5 places on each sides i.e 20 places total) leaving 8 places empty? I am getting answer as 12c5(selecting 5 numbers)*7c5(...
68 views

### Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
44 views

### Concerning The Number of Ways of Drawing a Full House vs. Two Pair

The Wikipedia entry for "Poker probability" gives the following result for the number of ways of drawing a full house: $$\binom{13}{1} \binom{4}{3} \binom{12}{1} \binom{4}{2} = 3, 744.$$ The logic ...
56 views

60 views

### Another form of Menage Problem : Place 8 more cherries(maroon) removing berries(black) 1 from each row and each column. No of ways?

I tried to see it as a matrix where for a position (i,j) , i+j = 8, 9, 16 means you can't change that position. Any help?
74 views

### Binomial coefficient paths?

Here's a problem and my attempt to answer it: We want to get a binomial coefficient identity depending on grid walking. Starting from the bottom left corner and going to the top right corner. You can ...
34 views

### What is the chance of randomly generating a given 10-character sentence? [closed]

Suppose we have an alphabet of the following allowed characters: the lowercase letters $a$ through $z$ (26) the uppercase letters $A$ through $Z$ (26) the numerals $0$ through $9$ (10) the common ...
64 views

The well-known Pólya-Vinogradov Inequality states: $$\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p,$$ where $\... 1answer 64 views ### Help in this little doubt in this proof from Hoffman and Kunze's Linear Algebra book I'm reading Hoffman and Kunze's Linear Algebra book and on page 177 they stated and proved the following theorem: It's a big proof which I didn't understand only a very little part of it: I ... 1answer 26 views ### How to find combinations with two conditions, one of them dependent on the other? How to find the number of combinatorial arrangements with two conditions, if one of the conditions is itself dependent on the second? The question below will make it clearer. The problem statement ... 2answers 95 views ### The number of positive integer solutions to the equation$x_1+x_2+…+x_n=n^2.$I'm working on this problem. To solve it I need this lemma: Let$n\ge2, n\in \mathbb N$. Let$X$be the number of solutions in positive integers to the equation$x_1+x_2+...+x_n=n^2$. Let$Y$be ... 1answer 19 views ### Expectation of absolute sum of squared normal distributions Let$u_i$be a standard normal distribution for all$i$. All$u_i$'s are independent of each other. I want to compute the expectation of: $$| \sum_i u_i^2 \lambda_i |$$ Where$\lambda_i$is real ... 1answer 44 views ### Expected number of couples having same number I have$n_1$red balls in a box$A$. These balls are numbered from$1, \cdots n_1$. Let make a copy version of box$A$, called box$D$(It means that the box$D$will contain$n_1$red balls from$1, \...
Thee Hamming distance $H(S_1, S_2)$ between two binary strings $S_1, S_2$ of length $n$ is the number of positions on which the two strings disagree. It is straightforward to show that if $S_1, S_2$ ...