Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Combinatorics is the study of finite or countable discrete structures — especially enumerative combinatorics: how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. This tag can be used for questions about permutations, combinations, partially ordered sets, bijection proofs, and generating functions.

Questions about:

These tags may be used alone or together with .

To learn more about enumerative combinatorics, see Wikipedia or Richard P. Stanley's Enumerative Combinatorics.

53034 questions
10
votes
2 answers

Let $x$ be an irrational number. Prove that there exist infinitely many rational numbers $\dfrac pq$ that satisfy the following

$$\bigg|\,x-\dfrac pq\,\bigg|<\dfrac 1{q^2+q}$$ My idea would be to solve the inequality for $\frac pq$ and then somehow use the pigeonhole principle. Is this heading in the right direction? Any hints would be helpful!
Sean Umlor
  • 111
  • 4
10
votes
2 answers

A polynomial sequence

I have a sequence of polynomials $Q_k(x, y)$, $k\geq 1$ defined recursively as follows: $Q_1=x$. There is a sequence of polynomials $p_j(y)$ of degree $j$ such that $Q_{2m}$ is of the…
No_way
  • 649
  • 4
  • 19
10
votes
2 answers

In how many ways can 1500 be resolved into two factors?

In how many ways can $1500$ be resolved into two factors? Is there a formula for that or a smart way because if I do that by listing all the divisors of $1500$ it will take a lot of time.
bulbasaur
  • 893
  • 7
  • 16
10
votes
1 answer

Bell numbers, recursions, bijections

The $n$th Bell number, named after Eric Temple Bell (although he was far from the first to think about them), is the number of partitions of a set of cardinality $n$. If they are written in the top row of this triangle, starting with the $0$th Bell…
Michael Hardy
  • 1
  • 30
  • 276
  • 565
10
votes
1 answer

What is the number of bijections between two multisets?

Let $P$ and $Q$ be two finite multisets of the same cardinality $n$. Question: How many bijections are there from $P$ to $Q$? I will define a bijection between $P$ and $Q$ as a multiset $\Phi \subseteq P \times Q$ satisfying the…
Douglas S. Stones
  • 20,413
  • 4
  • 66
  • 118
10
votes
1 answer

All clubs have a member among $n$ people

Let $n \geq 14$ be a positive integer. In a city there are more than $n$ clubs, all of them have exactly 14 members. At each group of $n+1$ clubs there is a person who is member of at least 15 of these $n+1$ clubs. Show that it is possible to…
jack
  • 1,236
  • 7
  • 20
10
votes
2 answers

A logic problem about set theory

In a group of n people, subgroups with common interest are formed (football,tennis,snooker). The number of subgroups equals $2^{n-1}$. Any 3 subgroups have a common member. Prove that there is a person who is a member of all the subgroups. proof by…
Maria Amarandei
  • 185
  • 6
10
votes
1 answer

Combinatorial interpretation of Delannoy numbers formula

The Delannoy number $D(a,b)$ can be defined as the numbers of paths on $\mathbb Z^2$ from $(0,0)$ to $(a,b)$ using only steps $(0,1)$, $(1,0)$ and $(1,1)$. It is straightforward to see that they follow the recursion (using either first-step or last…
FelixCQ
  • 1,258
  • 10
  • 14
10
votes
6 answers

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first $n$ natural numbers count the number of ways I can…
user238435
  • 103
  • 5
10
votes
4 answers

Number of ways you can form pairs with a group of people when certain people cannot be paired with each other.

Let's say you have a group of eight people and you want to form them into pairs for group projects. There are $\frac{8!}{4! 2^4}$ ways to do it. ($8!$ is the total number of ways $8$ people can be arranged in a line. Divide that by $2^4$, which…
Tara Roys
  • 235
  • 1
  • 2
  • 5
10
votes
2 answers

Counting number of moves on a grid

Imagine a two-dimensional grid consisting of 20 points along the x-axis and 10 points along the y-axis. Suppose the origin (0,0) is in the bottom-left corner and the point (20,10) is the top-right corner. A path on the grid consists of a series of…
Snowman
  • 2,586
  • 9
  • 35
  • 47
10
votes
2 answers

Is there a memorable solution to Kirkman's School Girl Problem?

Given a solution to Kirkman's School Girl Problem, it is of course easy enough to check that it actually is a solution. But how could you reconstruct it if you lost it? Is there a method or algorithm for constructing a solution which is easier to…
Lancel
  • 417
  • 4
  • 8
10
votes
4 answers

What is the proof of permutations of similar objects?

What is the proof of permutations of similar objects? I know the formula, but I cannot figure out how to derive it! permutations of similar objects The number of permutations of $n=n_1+n_2+\dots+n_r$ objects of which $n_1$ are of one type, $n_2$ are…
Rasa
  • 317
  • 1
  • 5
  • 16
10
votes
4 answers

How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?

I need to solve the problem, How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$? I've been given a hint, (Hint: Reduce the problem to $0 < j_1 < j_2 < . . . < j_k < n)$. The answer…
gsingh2011
  • 749
  • 1
  • 8
  • 23
10
votes
3 answers

Closed form of factorial and cascading power sum

Consider the following sum: $$ \sum_{i =0}^{j} \left( \frac{(j-i)^ix^i \ln(x)^{(j-i)}\ln(x)^i}{(j-i)!i!} \right) $$ I can simplify the sum to: $$ \ln(x)^j\sum_{i =0}^{j} \left( \frac{(j-i)^ix^i}{(j-i)!i!} \right) $$ Furthermore I can observe that…
Sidharth Ghoshal
  • 14,653
  • 8
  • 33
  • 81
1 2 3
…
99
100