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.

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The number of ordered pairs of positive integers $(a,b)$ such that LCM of a and b is $2^{3}5^{7}11^{13}$

I started by taking two numbers such as $2^{2}5^{7}11^{13}$ and $2^{3}5^{7}11^{13}$. The LCM of those two numbers is $2^{3}5^{7}11^{13}$. Similarly, If I take two numbers like ...
77 views

Does an Eulerian semi-graceful polyhedral graph exist?

In a graceful graph, the vertices have number values that range from 0 to $n$ and $n$ edges with all values from 1 to $n$ that are differences between the vertex values. Here's a graceful but boring ...
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Each number in a subset $S\subseteq \{1,\ldots,2n\}$ does not divide another one. Then $\max |S|$?

This problem comes from a seemingly innocuous question from a professor during a lesson for a Math Olympiad course. [A part of this question is really a classic of number theory/combinatorics] ...
153 views

Improvised Question: Combination of selection of pens

This is a improvised version of the question here. Supposing there are four brands of pens, W, X, Y, Z. You want to choose $10$ pens made up of any combination of the brands, but limited to a ...
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Odd prime combinatorics problem [closed]

How should I show that ${2p \choose p}\equiv 2\pmod p$ if p is an odd prime! help please
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probability that 5 square lie along a diagonal line - doubt [duplicate]

If 5 squares are chosen at random from a chess board, what is the probability that they lie on a diagonal line? this is the same question indeed. Answer is given by Mr.Brian M. Scott. But I got a ...
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How many degree m elements in the exterior algebra on n generators over a finite field, vanish when raised to the r-th power?

Let $R=\Lambda_{\mathbb{F}_p}[e_1,...,e_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements (this arises naturally as the mod-p cohomology ring of the $n$-dimensional ...
6k views

How do I determine the number of odd integers in a range?

Say I have a range of integers, [1..100]. How do I determine the number of odd integers in that range? Does it make a difference if the beginning and end numbers are [odd .. odd], [even .. odd], [even ...
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Finding coefficients of $x^n$ and $x^{n+r}$ in an expansion

I have to find the coefficients of $x^n$ and $x^{n+r}$ $(1 < r < n)$ in the expansion of: $$(1 + x)^{2n} + x(1 + x)^{2n - 1} + x^2(1 + x)^{2n - 2} + ... + x^n(1 + x)^n$$ How do I solve it?
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Paths starting from a given node that touch each node a given number of times

How many paths starting from a given node touch each node a given number of times? We have a complete graph with vertices $1,2,3…j$. We want to know the number of paths of length $N$, starting from ...
1k views

Combinatorics-permutations and combinations

In a soccer tournament of 15 teams, the top three teams are awarded gold, silver, and bronze cups, and the last three teams are dropped to a lower league. We regard two outcomes of the tournament as ...
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Does there exist integer such that there exist sum of powers congruent mod $p$?

Let $n \in \mathbb{N}$, $p$ prime. For arbitrary $C \in \mathbb{Z}$, does there exist $a_1, a_2, \dots, a_n \in \mathbb{Z}$ such that$$C \equiv \sum_{i=1}^n a_i^n \text{ }(\text{mod }p)?$$
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Hamiltonian cycles in associahedron graphs

Let two distinct fully parenthesized products of $n$ symbols be called adjacent provided one of them may be obtained from the other by a single application of the associative law. Such graphs may be ...
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Counting the size of the largest sets of independent strings

This question derives from a PPCG coding challenge I posed previously. For a given positive integer $n$, consider all binary strings of length $2n-1$. For a given string $S$, let $L$ be an array of ...
532 views

points with integer coordinates inside triangles in $\Bbb{R}^3$

I'm taking off from Number of lattice points inside a triangle and its area Consider a triangle in $\Bbb{R}^3$ whose vertices have integer coordinates. What's the fastest way of counting how many ...
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How many unique numbers can be obtained by adding two numbers from two different sequences?

Let the two integer sequences $\{a_m\}$ and $\{b_m\}$, be defined as: $a_n+D_n=a_{n+1}$ and $b_n=a_n-k$, where $D_n$ may be any natural number (and $D_i$ may or may not be equal to $D_j$), $k$ is an ...
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Let $g_{n}$ be the no. of derangements with $n$ elements and $f_{n}$ the no. of permutations with one fixed point. Show that $|g_{n}-f_{n}|=1$

This is a problem from Loren Larson's "Problem solving through problems", 2.5.13, page 78. Let $S_{n}=${$1,2,...,n$}. A derangement of $S_{n}$ is a permutation with no fixed points. Let $g_{n}$ be ...
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Combinatorics - Counting the number of binary strings with specified length and sum, with substring constraints

Suppose I have a string of bits of length R. The sum of the bits must be equal to S, so there are S ones and R-S zeros. The longest string of ones cannot exceed X in length. Also the number of places ...
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calculation of all possible combinations.

Suppose we are given $x_1 - x_2 = 31$. Constraints - $0 \leq x_1 \leq 45$ and $0 \leq x_2 \leq 45$. Then we have to tell number of all possible distributions for $x_1$ and $x_2$.
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What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial ...
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Discrete mathematics: Question regarding “Pigeonhole principle”. [closed]

Each point in the plane is coloured either red or blue. Show that there are two points of the same colour which are exactly 1 cm apart.
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