# 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.

4,484 questions
Filter by
Sorted by
Tagged with
48k views

5k views

### Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$

I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
40k 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 ...
34k 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 $\left\{1,1,2,3,5,8,13,21,...\right\}$.
21k 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 ...
7k views

### 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 ...
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 ...
6k 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 ...
10k views

### How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$?

How can we prove that $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$ (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.
5k views

### why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$? I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward. ...
34k views

91k 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 ...
2k views

### 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 ...
51k 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 ...
8k 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 ...
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 ...
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?
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 ...
525 views

### How many $N$ digits binary numbers can be formed where $0$ is not repeated
How many $N$ digits binary numbers can be formed where $0$ is not repeated. Note - first digit can be $0$. I am more interested on the thought process to solve such problems, and not just the answer. ...
I came across this formula in a list: The number of ways of distributing $n$ identical objects among $r$ groups such that each group can have $0$ or more $(\le n)$ objects I know that standard way ...