# 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.

4,235 questions
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1answer
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### Probability distribution in the coupon collector's problem

I'm trying to solve the well known Coupon Collector's Problem by explicitly finding the probability distribution (so far all the methods I read involve using some sort of trick). However, I'm not ...
4answers
32k views

### The generating function for the Fibonacci numbers

Prove that $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$ The coefficients are Fibonacci numbers, i.e., the sequence $\left\{1,1,2,3,5,8,13,21,...\right\}$.
7answers
20k views

### Making Change for a Dollar (and other number partitioning problems)

I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted ...
1answer
9k views

### 6-letter permutations in MISSISSIPPI

How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller ...
3answers
37k views

### How to use stars and bars?

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$, where $x_i\in\mathbb{N}$. Is this the correct time to apply the method? This is ...
3answers
5k views

### Number of permutations of $n$ where no number $i$ is in position $i$

I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is ...
3answers
33k views

6answers
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### Number of occurrences of k consecutive 1's in a binary string of length n (containing only 1's and 0's)

Say a sequence $\{X_1, X_2,\ldots ,X_n\}$ is given, where $X_p$ is either one or zero ($0 < p < n$). How can I determine the number of strings, which do contain at least one occurrence of ...
5answers
7k views

### How do you prove ${n \choose k}$ is maximum when $k$ is $\lceil \tfrac n2 \rceil$ or $\lfloor \tfrac n2\rfloor$?

How do you prove $n \choose k$ is maximum when $k$ is $\lceil n/2 \rceil$ or $\lfloor n/2 \rfloor$? This link provides a proof of sorts but it is not satisfying. From what I understand, it focuses on ...
5answers
84k views

### Number of onto functions

What are the number of onto functions from a set $\Bbb A$ containing m elements to a set $\Bbb B$ containing n elements. I ...
7answers
49k views

### In how many ways can a number be expressed as a sum of consecutive numbers?

All the positive numbers can be expressed as a sum of one, two or more consecutive positive integers. For example $9$ can be expressed in three such ways, $2+3+4$, $4+5$ or simply $9$. In how many ...
1answer
2k views

### Find the number of arrangements of $k \mbox{ }1'$s, $k \mbox{ }2'$s, $\cdots, k \mbox{ }n'$s - total $kn$ cards.

Find the number of arrangements of $k \mbox{ }1'$s, $k \mbox{ }2'$s, $\cdots, k \mbox{ }n'$s - total $kn$ cards - so that no same numbers appear consecutively. For $k=2$ we can compute it by using ...
2answers
3k views

### Number of monomials of certain degree

Wikipedia says that the number of different monomials of degree $M$ in $N$ variables is $$\frac{(M+N-1)!}{M!(N-1)!}\; .$$ Can anyone explain why this is true?
5answers
3k views

### How many ways can $b$ balls be distributed in $c$ containers with no more than $n$ balls in any given container?

I think there are $\binom{b+ c - 1}{c-1}$ ways to distribute $b$ balls in $c$ containers. (Please correct me if that's a mistake.) How does this change if I am not allowed to place more than $n$ balls ...
3answers
3k views

### Proof of a combinatorial identity: $\sum\limits_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$ [duplicate]

Possible Duplicate: Identity involving binomial coefficients This was part of a homework assignment that I had, and I couldn't figure it out. Now it is bugging me. Can anyone help me? Although a ...
2answers
502 views

2answers
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### Distinct ways to keep N balls into K Boxes?

How many different ways I can keep N balls into K boxes, where each box should atleast contain ...
3answers
2k views

### How to prove that the number $1!+2!+3!+…+n!$ is never square?

How to prove that the number $1!+2!+3!+...+n! \ \forall n \geq 4$ is never square? I was told to count permutations but I cannot figure out what we are permuting.... Thanks!