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.

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6 views

How changing parameters on a sigma notation affects your position on Pascal's Triangle

Below is a proof claiming to prove that the sum of a row on Pascal's Triangle is equal to 2x the sum of the row before it. The key here was to use Pascal's Identity. I understand the general concept, ...
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1answer
22 views

counting proof for $\sum$

I was hoping someone could help me come up with a combinatorial proof, one with an easy to understand 'story' for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$. For instance I understand that $2^n$ is the number ...
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1answer
9 views

ice cream flavors, cones, and toppings. A counting question.

There are 5 distinct ice cream flavors, 5 distinct cones, and 5 distinct toppings. I gather all 15 distinct supplies. There are 5 people. Each person has to get a cone, flavor, and topping, and I can ...
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2answers
18 views

Symmetric random walk that does not hit $0$

Suppose a random walk $\{X_n\}_{n\in \mathbb Z^{\ge 0}}$ on the integer number line, $ \mathbb {Z}$ , which starts at $0$ and at each step moves $+1$ or $−1$ with equal probability. The increments ...
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1answer
21 views

Counting number of valid strings

Call a length $n$ string "valid" if it is formed from the set $\{A, B, C, D, E, F, G\}$ and it contains at least one of A, B, C, D Find the number of valid strings using Principle of Inclusion-...
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1answer
24 views

How many binary strings of length $n$ contain $k$ flips?

If I have say the string $1010010001010101$, which has a length of $16$ and there are $12$ flips. My thoughts are to just count the number of ways I can stick a $10$ in the there so ${n-1 \choose 0.5k}...
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1answer
17 views

Counting distinct functions

I would like to get help with the following question: A and B are integers (A ≤ B). How many distinct functions exist from the type of [A...B] → [A..B] such that f(x) ≤ x for all x in the ...
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2answers
35 views

How to calculate the complement of rolling dies?

I'm having a hard time trying to picturing the complement rule on this one. Can someone please enlighten me? You roll 10 times of a fair 12 sided die. Find the probability of rolling two or more 12's ...
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0answers
15 views

Why is $\mathcal{P}$ = SET($\mathcal{Z}$)?

in the book Analytic Combinatorics, where the symbolic method is described, they say that $\mathcal{P}$ = $SET(\mathcal{Z})$ and therefore $\hat{P}(z) = exp(\hat{Z}(z))$. $\mathcal{P}$ denotes the ...
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1answer
21 views

Generating functions (with symbol method) of special partitions of natural numbers

I want to show that the generating function of the number of partitions where every summand appears at most twice of every natural number n equals the number of paritions of n into summands which are ...
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1answer
19 views

Number of steps in special graph algorithm.

First of all to avoid misunderstanding i am going to describe what will i call a algorithm in my graph-theoretic problem. algorithm is a set of pairs, where first coordinate is natural number - a ...
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1answer
20 views

Number of possible ways n people can meet for lunch on a circular table such that each day no two people sit together who have sat together before?

So i started studying this book on graph theory by Nar Singh Deo, and fairly early on the following problem gets introduced: n people decide to meet for lunch every day, however they decide that they ...
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0answers
11 views

Selecting 'r' objects from different groups of alike objects

From 4 red, 3 black and 2 white balls, in how many ways can one select 3 balls? Also answer for the case when at least one black ball must be selected.
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1answer
44 views

Proving a square number

a) Decide (without the help of a calculator) which of the following numbers $t = 1 125$, $u = 1 225$, $v = 111 225$ and $w = 112 225$ square numbers. If necessary, specify the numbers whose squares ...
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3answers
252 views

There are 51 natural numbers between 1-100, proof that there are 2 numbers such that the difference between them equals to 5

This question relates to pigeonhole principle, but I couldn't find what the holes need to be. How should I prove this сlaim?
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2answers
43 views

A question on combinatorics: if $p + q + r = 10$, find the number of possible combinations where the condition holds true.

If $ p + q + r = 10 $, find the number of possible combinations satisfying the equation. ( $p,q,r ≥ 0$) PS - As suggested by Brian Moehring, I used the 'stars and bars' method. We need the sum as $...
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1answer
19 views

Maximizing probability of winning with urn of 30 balls, 20 white and 10 black. white gives +1, black gives -2

An urn contains 20 white balls and 10 black balls. If you draw a white ball, you get 1 dollar, but if you draw a black ball, you lose 2 dollars. How many balls should you draw in order to maximize ...
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1answer
24 views

Number of ways in which 3 numbers in Arithmetic progression can be selected from 1,2,3…n is?

Initially i thought of dividing the sequence into groups of 3 that is 1,2,3,4,.. gets divided in to consecutive groups which means there are a total of $n\over 3$ groups and hence the total was of ...
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1answer
20 views

Arrangements around a square, People not sit across diagonally - Extension

The first two parts of this question are exact same as the Counting the arrangements of 8 people around a square table? and have already been answered but my struggle is in the final part of my ...
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1answer
20 views

Ice cream, cones, and toppings. How many ways can I make the order?

There are five orders of ice cream. Each ice cream order has a distinct cone, distinct ice cream flavor, and distinct topping. I have to put on a cone before I can put on ice cream, and I have to put ...
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0answers
16 views

How to prove following identity in measure of integral?

We have a point $P$ in $\mathbb{R}^2$ having coordinates $(x_+,x_-)$. Let $\xi$ be a path from the origin to $P$ consisting of polygonal path parallel to $x$ and $y$ axis. Let $l_i^+$ denote the ...
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0answers
7 views

Real World Application of Dominating Sets and Total Dominating Sets [on hold]

Are there any basic applications of dominating sets and total dominating sets?
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2answers
32 views

Counting with Venn Diagrams

How many arrangements are there of MURMUR with no pair of consecutive letters are the same. This problem is from Alan Tucker's book: applied combinatorics.
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2answers
53 views

Number of words with two “$A$” using two letters from “$RATA$” and three letters from “$TIERRA$”

Find the number of words with two "$A$" using two letters from "$RATA$" and three letters from "$TIERRA$". What I did: There are two cases, one where I choose both $A$ from $RATA$ and the rest of ...
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0answers
42 views

How many five digit numbers can be formed from the integers 1,2,..,9 with one digit appearing at max thrice?

The problem is in the title of the question, repeated below: How many five digit numbers can be formed from the integers 1,2,..,9 with one digit appearing at max thrice? I have two solutions as ...
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1answer
34 views

In how many ways can $n$ unique books be arranged on $m$ shelves if at least one shelf is empty?

In how many ways can $n$ unique books be arranged on $m$ shelves if at least one shelf is empty? I am pretty sure that this is a question asking about objects and dividers, but I'm not sure how to ...
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0answers
27 views

Need help with Möbius function

Suppose I need to find total subset of numbers of length $K$ in range $1$ to $N$ such that their $\gcd$ is $g$. How can I utilize Möbius function for that. So approach is we can choose $K$ numbers ...
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2answers
30 views

L-stones with quadrat [on hold]

Figure A 591014 a gives a square of 25 fields. This figure should be designed with (congruent) L-stones, whereby a special field is removed beforehand. An L-shaped stone consists of three square ...
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1answer
42 views

Probability of balls in urn

For a random experiment, four red and four blue balls are available. In the beginning, four of the eight bullets are placed in one urn, the other four serve as a supply. In each step of the random ...
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2answers
30 views

Three different numbers from the set {1, 2, 3, 4, 5, 6}. In how many ways can he do this so that the three numbers are not consecutive?

Three different numbers from the set {1, 2, 3, 4, 5, 6}. In how many ways can he do this so that the three numbers are not consecutive? So it's C (6,3)=20 There are 4 types that are consecutive (1,2,...
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1answer
12 views

How many different combinations can be put to decorate the room? [on hold]

Suppose there are 4 identical dolls, 6 identical toys, 9 identical pillows and we can use any amount to decorate a room. In how many different ways can you decorate the room given that the room cannot ...
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1answer
19 views

How much 3x4 matrix with integer numbers (every num >= 0) , which in every row the sum is 3 , and there no column of zero's?

I tried to answer the question and get a result of 7998 options , that is right? from my way of solution , there are 20^3 3x4 matrixes which sum of every row is 3, and then i tried to substract num ...
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0answers
22 views

Forming n digit numbers out of 1,2,…,9 such that k digits can repeat

I came across following two problems: (1) How many 5-digit numbers can be formed from the integers 1,2,…,9 with only one digit appearing twice? (2) How many 5-digit numbers can be formed from ...
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3answers
64 views

How many solutions does the rebus have? [on hold]

How many 8 digit (its digits from left to right are labeled a through g) satisfy the following constraints: $a_1 < a_2 < a_3 < a_4$ $a_4 > a_5$
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3answers
66 views

Symmetric matrix as a sum of symmetric matrices

Let matrix $M \in \mathbb{N}^{5 \times 5}$ be symmetric with non-negative integer entries and zeros on the main diagonal and having the property that the row sums are equal to $2r$ for some $r \geq 2$....
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1answer
25 views

Grouping Combination Probability

$9$ buses are being sent to $3$ airports such that $3$ go to Airport $1$, $5$ go to Airport $2$, and $1$ goes to Airport $3$. Assume that the buses are sent to the airports at random. $1$) In how ...
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2answers
26 views

Permutation allowing repetition

Jeff is going to eat dinner out each day from Monday to Friday in a certain week, with each dinner being at one of his $15$ favorite restaurants. $1$) How many possibilities are there for Jeff's ...
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4answers
56 views

A question on giving prizes when there is no restriction on the number of prizes per person

A group consisting of $3$ men and $6$ women attends a prizegiving ceremony. If $ 5$ prizes are awarded at random to members of the group, find the probability that exactly $3 $ of the prizes are ...
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0answers
15 views

Combination-Counting problem [duplicate]

How many different ways can 200 coins be distributed among 10 people so that: 1)everyone gets 0, or more, coins? 2)everyone gets 5, or more, coins? I know how to do question it they have an exact ...
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0answers
16 views

Generalized Gessel-Viennot-Lindstrom Lemma to Hyperdeterminants of Multinomial Coefficients

A consequence of the Gessel-Viennot-Lindstrom lemma is that if $A$ and $B$ are collections of points in $\mathbb{Z}^2$, that the number of non-intersecting lattice paths (counted with a sign) from the ...
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1answer
54 views

A combinatorics equality(need help proving it)

show that $\binom{n-1}{\frac{1}{2}(n+b)-1}-\binom{n-1}{\frac{1}{2}(n+b)}=\binom{n}{\frac{1}{2}(b+n)}\frac{b}{n}$. Assuming $b>0$ and $b<n$ Just by thinking combinatorically, I don't see why the ...
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2answers
48 views

Which is the importance of Young’s tableaux in mathematics?

I don’t know much about combinatorics, I’m just getting started on this. I want to know, why Young’s tableaux are important? and why it is important to relate them to matrices? Thank you very much.
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1answer
51 views

To Prove $ |Y_{n,k}|=(n/k)*{n-2k-1 \choose k-1}$

To prove, $$ |Y_{n,k}|=\frac{n}{k}{n-2k-1 \choose k-1}$$ Where, For $k,n \in N$, let $Y_{n,k}$ be the collection of $k-element$ subsets $A \subset [n]$ , given that, $i-j$ is not congruent to 1 or ...
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0answers
20 views

How many ways can balls be picked from a box? [duplicate]

Suppose you have $a$ orange, $b$ pink, and $c$ purple balls in a box. How many ways can 3, where $c < a, b, c$ balls be selected such that at least one of the three is purple? I think you just ...
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0answers
42 views

Drawing balls from $n$ numbered balls $k+1$ times with replacement

Consider: $E_j$: the maximum of the values obtained on $k+1$ draws is $j$, $j=1,\ldots…,n$, $F_j$: all the draws are done on $\{1,\ldots,j\}$, $j=1,\ldots,n$ $F_0 = \emptyset$. Calculate P($E_j$)...
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2answers
33 views

Find the number of four-letter words that use letters from {A, B, C} in which no three consecutive letters are the same.

Case 1: (3)(3)(2)(3)=54. In the first two spots the letters can be any of A,B, or C. The third spot is where we risk having a third consecutive letter, so there are only 2 choices. Now the fourth spot ...
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0answers
24 views

Accessible sources providing an intuitive, elementary intro to Hopf algebras/monoids

Motivation: I'm interested in understanding the role that noncrossing partitions play in Hopf algebras/monoids (HAs) as the components of the power series of the compositional inverse of formal ...
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1answer
80 views

Prove Pascal's formula by induction [on hold]

I have Pascal's identity: $$\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}.$$ How can I prove this using mathematical induction? This is an exercise from Section 5.5 of Alan Tucker's Applied ...
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0answers
19 views

Statistics question about different combinations [on hold]

Consider a 6 vs 6 youth soccer team (i.e., teams field 6 players at once) with 9 players. (a) How many different combinations of players can the team field? (b) If the team has exactly one goalie (who ...
3
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4answers
126 views

How many integral solutions does $2x + 3y + 5z = 900$ have when $ x, y, z \ge 0$?

Solution: Let $2x + 3y = u.$ Then we must solve $\begin{align} u + 5z = 900 \tag 1 \\ 2x + 3y = u \tag 2 \end{align}$ For $(1),$ a particular solution is $(u_0, z_0) = (0, 180).$ Hence, all the ...