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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6
votes
4answers
666 views

Prove $\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) [duplicate]

Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?
26
votes
4answers
19k views

Combinatorial proof of summation of $\sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$

Can you guys help me prove this? There is a way of proving this logically but I was hoping to find a more "mathematical" proof, if possible. $$\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n ...
47
votes
2answers
4k views

Identity for convolution of central binomial coefficients

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
47
votes
9answers
21k views

Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to ...
14
votes
9answers
8k views

Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$

I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= ...
31
votes
9answers
2k 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. ...
12
votes
3answers
6k views

I have a problem understanding the proof of Rencontres numbers (Derangements)

I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation $$D_{n,0}=\left[\frac{n!}{e}\right]$$ where $[\cdot]$ denotes the rounding function (i.e., ...
40
votes
4answers
8k views

Probability for the length of the longest run in $n$ Bernoulli trials

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, ...
12
votes
4answers
2k 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 ...
6
votes
5answers
2k views

Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

I tried to solve it using induction, but that got me no were, in the middle of the equation stat appearing ks that I don't see how to get out of the equation. I think the easiest way to prove it, it's ...
23
votes
4answers
13k 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 $\left\{1,1,2,3,5,8,13,21,...\right\}$.
19
votes
7answers
10k 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 ...
6
votes
3answers
2k views

Enumerating number of solutions to an equation

How do you find the number of solutions like this? $$x_1 + x_2 + x_3 + x_4 = 32$$ where $0 \le x_i \le 10$. What's the generalized approach for it?
9
votes
4answers
1k views

Combinatorial proof

Using notion of derivative of functions from Taylor formula follow that $$e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ Is there any elementary combinatorial proof of this formula here is my proof for ...
9
votes
3answers
1k 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 ...
7
votes
3answers
2k views

Proof of a combinatorial identity: $\sum_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$ [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? ...
14
votes
1answer
5k 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 ...
1
vote
1answer
877 views

extended stars-and-bars problem(where the upper limit of the variable is bounded)

The problem of counting the solutions $(a_1,a_2,\ldots,a_n)$ with integer $a_i\geq0$ for $i\in\{1,2,\ldots,n\}$ such that $$a_1+a_2+a_3+....a_n=N$$ can be solved with a stars-and-bars argument. What ...
10
votes
1answer
1k views

In how many ways can we colour $n$ baskets with $r$ colours?

In how many ways can we colour $n$ baskets using up to $r$ colours such that no two consecutive baskets have the same colour and the first and the last baskets also have different colours? For ...
28
votes
1answer
3k views

How to reverse the $n$ choose $k$ formula?

If I want to find how many possible ways there are to choose k out of n elements I know you can use the simple formula below: $$ \binom{n}{k} = \frac{n! }{ k!(n-k)! } .$$ What if I want to go the ...
4
votes
3answers
2k views

Demonstrate another way to solve the Inclusion–exclusion principle?

I'm attempting to solving the Inclusion–exclusion principle, which is generally described as follows... $$\begin{align} \left| \bigcup\limits_{i=1}^n A_i \right| = + \left( \sum\limits_{i=0}^n | ...
8
votes
2answers
5k views

Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$

I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$ I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively ...
3
votes
1answer
649 views

How can I determine the number of unique hands of size H for a given deck of cards?

I'm working on a card game, which uses a non-standard deck of cards. Since I'm still tweaking the layout of the deck, I've been using variables as follows: Hand size: $H$ Number of suits: $S$ Number ...
10
votes
5answers
455 views

What does $\binom{-n}{k}$ mean?

For positive integers $n$ and $k$, what is the meaning of $\binom{-n}{k}$? Specifically, are there any combinatorial interpretations for it? Edit: I just came across Daniel Loeb, Sets with a ...
5
votes
2answers
892 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?
29
votes
2answers
5k views

How do I count the subsets of a set whose number of elements is divisible by 3? 4?

Let $S$ be a set of size $n$. There is an easy way to count the number of subsets with an even number of elements. Algebraically, it comes from the fact that $\displaystyle \sum_{k=0}^{n} {n ...
14
votes
5answers
13k views

Calculating the total number of surjective functions

It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ...
4
votes
1answer
616 views

Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?

A known identity of binomial coefficients is that $$ \sum_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}. $$ Is there a combinatorial proof/explanation of why it holds? Thanks.
3
votes
2answers
185 views

Sum of combinations with varying $n$ [duplicate]

What is the sum of number of ways of choosing $n$ elements from $(n+r)$ elements where $r$ is fixed and $n$ varies from $1$ to $m$ ? Can this be reduced to a formula ? $$ \sum ^m _{n=1} \binom{n + ...
6
votes
2answers
257 views

Sum of Stirling numbers of both kinds

Let $a_k$ be the number of ways to partition a set of $n$ elements $orderly$,which means that order of subsets matters, but order of elements in each subset does not. My task: Prove, ...
5
votes
5answers
2k 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 ...
5
votes
2answers
3k views

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 ...
11
votes
3answers
7k views

Simplifying Catalan number recurrence relation

While solving a problem, I reduced it in the form of the following recurrence relation. $ C_{0} = 1, C_{n} = \displaystyle\sum_{i=0}^{n - 1} C_{i}C_{n - i - 1} $ However ...
8
votes
2answers
18k views

Unique ways to keep N balls into K Boxes?

How many unique ways I can keep N balls into K boxes, where each box should atleast contain ...
6
votes
5answers
4k views

How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
3
votes
6answers
374 views

Probability problem

I have $3$ coins, $1$ coin has $2$ heads (HH), 1 coin has $2$ tails (TT), $1$ coin has $1$ head and $1$ tail (HT). I toss the coin, it fells on my hand, and the side i see is a tail. What's the chance ...
17
votes
1answer
922 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 ...
13
votes
3answers
7k views

Combinations of selecting n objects with k different types

Suppose that I am buying cakes for a party. There are k different types and I intend to buy a total of n cakes. How many different combinations of cakes could I possibly bring to the party?
5
votes
4answers
304 views

Proving $\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$

Prove that $$\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$$ by computing the coefficient of $z^M$ in the identity $$(1 + z + z^2 + \cdots ) \cdot \frac{1}{(1-z)^{k+1}} = \frac1{(1-z)^{k+2}}.$$ I ...
3
votes
1answer
238 views

If $|A| > \frac{|G|}{2} $ then $AA = G $ [closed]

I'v found this proposition. If $G$ is a finite group , $ A \subset G $ a subset and $|A| > \frac{|G|}{2} $ then $AA = G $. Why this is true ?
2
votes
2answers
215 views

Double Factorial: Number of possibilities to partition a set of $2n$ items into $n$ pairs

I know that the partition of $2n$ items into $n$ pairs has something to do with double factorial, but I am not sure how many possibilities we exactly have. We can choose such a partition into pairs ...
23
votes
1answer
5k views

How to count number of bases and subspaces of a given dimension in a vector space over a finite field?

Let $V_{n}(F)$ be a vector space over the field $F=\mathbb Z_{p}$ with $\dim V_{n} = n$, i.e., the cardinality of $V_{n}(\mathbb Z_{p}) = p^{n}$. What is a general criterion to find the number of ...
26
votes
7answers
20k 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 9. In how many ways can a number ...
9
votes
6answers
788 views

Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
16
votes
2answers
1k views

What's the General Expression For Probability of a Failed Gift Exchange Draw

My family does a gift exchange every year at Christmas. There are five couples and we draw names from a hat. If a person draws their own name, or the name of their spouse, all the names go back in a ...
13
votes
5answers
47k views

Number of ways of distributing n identical objects among r groups

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 ...
6
votes
5answers
12k views

Number of relations that are both symmetric and reflexive

Consider a non-empty set A containing n objects. How many relations on A are both symmetric and reflexive? The answer to this is $2^p$ where $p=$ $n \choose 2$. However, I dont understand why this is ...
14
votes
3answers
2k views

Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.

Prove that $$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$ I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its ...
1
vote
5answers
721 views

Number of solution for $xy +yz + zx = N$

Is there a way to find number of "different" solutions to the equation $xy +yz + zx = N$, given the value of $N$. Note: $x,y,z$ can have only non-negative values.
5
votes
3answers
782 views

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

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac{n}{2} \rceil $ or $ \lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...