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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1
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0answers
18 views

Tennis league - minimum number of games

In a tennis league, consisting of 32 players, each player played with each other at least twice and at maximum Q times. Knowing that each of them participated in a different number of games, can you ...
2
votes
1answer
744 views

Probability of finding specific set of coloured balls within larger set of random-drawn balls

In this question I was helped with calculating the probability of drawing specific set of M coloured balls from a set of N coloured balls. Now I am looking for a solution for an extended problem: ...
2
votes
1answer
18 views

Calculating the number of elements of a given order in a group of permutations.

Let $S$ denote the group of all those permutations of the English alphabet that fix the letters T, E, N, D, U, L, K, A, and R. Other letters may or may not be fixed. Show that $S$ has elements ...
2
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1answer
63 views

Number of ways in which papers can be arranged so that mathematics papers do not come together?

$11$ papers are set for an examination in which two are of mathematics. Number of ways in which papers can be arranged so that mathematics papers do not come together? There are $11!$ ways in total. ...
5
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3answers
116 views

Why it has to be an integer?

Let $k$ and $n$ be integers greater than 1. Then $(kn)!$ is not necessarily divisible by A. $(n!)^k$ B. $(k!)^n$ C. $n!\cdot k!$ D. $2^{kn}$ I believe option D is correct ...
0
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0answers
20 views

isomorphic permutation groups with same cycle index [on hold]

Is there two nonidentical isomorphic permutation groups with same cycle index? (Two permutation groups A and B on sets X and Y, respectively, are said to be identical, if there is a function 1-1 map ...
0
votes
1answer
10 views

Existence of a $d$-regular graph such that $|N_G(x) \cap N_G(y)| = \lambda$.

Consider a $d$-regular graph $G = (V, E)$ of order $n$ such that $|N_G(x) \cap N_G(y)| = \lambda$ for all distinct $x, y \in V$. By double counting we have a necessary condition $\lambda (n - 1) = d(d ...
1
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3answers
19 views

How many ways can you pick out 15 candies total to throw unordered into a bag and take home

A store sells 8 kinds of candy. How many ways can you pick out 15 candies total to throw unordered into a bag and take home. here 15 candies.. so we choose 8 from out of 15 is ..=$^{15}C_8$ is i am ...
1
vote
1answer
31 views

Vandermonde's Convolution special case.

I am not able to show this case of Vandermonde's Convolution without using induction. Can someone help me? $$ \binom{n}{m} = \sum_{k=0}^{m} \binom{n-p}{m-k} \binom{p}{k}. $$ I thank now.
1
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2answers
44 views

What is the probability to fill rows of a cinema hall?

This is the problem I'm trying to solve, but I'm not sure I'm on the correct path! would appreciate your feedback guidence and help. So the problem is: there're 3 rows in a cinema hall. the first one ...
1
vote
0answers
38 views

Find the number of ways to select six distinct integers from the set $\{1,2,\dotsc,49\}$ such that no two consecutive integers are selected. [on hold]

Find the number of ways to select six distinct integers from the set $\{1,2,\dotsc,49\}$ such that no two consecutive integers are selected.
4
votes
4answers
256 views

Sum of sum of binomial coefficients

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
33
votes
2answers
619 views

Placing the integers $\{1,2,\ldots,n\}$ on a circle ( for $n>1$) in some special order

For which integer $n>1$ can we place the integers $\{1,2,\ldots,n\}$ on a circle (say boundary of $S^1$ ) in some order such that for each $s \in \{1,2,\ldots,\dfrac {n(n+1)}{2}\}$ , there exist a ...
0
votes
1answer
18 views

Does every partition of n correspond to some permutation of [1,2, … n]?

It is known that every permutation can be decomposed into disjoint cycles. The cycle type gives the length of each cycle. The sum of cycles length is n. I am wondering whether every partition of n ...
1
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0answers
23 views

Is there a combinatorial identity for the following sum?

Let $a,b,c$ be integers. Is there by any chance a neat combinatorial identity for the following sum? $$ \sum_{j=0}^c{a + jb \choose j}. $$ Thanks!
0
votes
2answers
29 views

Probability that digits 1,2 and 3 will appear in a decimal m digits, how do I tweak my thinking to be correct?

So I first thought to approach this as the complement of an inclusion exclusion problem, $P(A_{1}\cap A_{2}\cap A_{3})=1-P(A_{1}\cup A_{2}\cup A_{3})$ Where $A_{i}$ is the event that digit i appears ...
2
votes
0answers
70 views

Number of games required such that two arbitrary players play together and against each at least once.

There are $2N$ players to form two teams of $N$ players that play against each other in a game. How many games are required such that two arbitrary players play together and against each other at ...
0
votes
0answers
24 views

How many 4-permutations on {1.. 9} are there? [on hold]

Please if you could explain how to solve this problem. How many 4-permutations on {1.. 9} are there?
3
votes
3answers
44 views

How many $5$-digit numbers (including leading $0$'s) are there with no digit appearing exactly $2$ times?

How many $5$-digit numbers (including leading $0$'s) are there with no digit appearing exactly $2$ times? The solution is supposed to be derived using Inclusion-Exclusion. Here is my attempt at a ...
0
votes
1answer
17 views

r-combination from n objects where objects can be indistinguishable or distinguishable

How to solve this kind of combinatorial problem. You are given n objects and you have to find out r-combination from it. As example there are 4 objects.. 1 2 2 3..you have to find out how many ...
7
votes
2answers
99 views

How can I get f(x) from its Maclaurin series?

I know how to get a Maclaurin series when $f(x)$ is given. I have to find $\sum_{n=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k$. But how can I get $f(x)$ from its Taylor series? The problem is $$f(x) = ...
-4
votes
2answers
47 views

How many $4$-sequences on $\{1, \ldots, 9\}$ are there? [on hold]

How many $4$-sequences on $\{1, \ldots, 9\}$ are there? Please explain how to solve this problem as I am lost, as the text book does not explain that well.
3
votes
2answers
39 views

Combination of elements in a ring and selecting non adjacents

Ok, suppose we have a clock, with the usual design of numbers ordered from 1 to 12 (so 1 and 12 are adjacents). The question is what is the number of possible combinations of four non adjacent ...
2
votes
1answer
48 views

How many more edges can be added to a graph while keeping it acyclic?

If I have a connected, directed graph with $n$ vertices and $m$ edges, is there some sort of formula that describes how many more edges can be added to the graph while keeping it acyclic?
14
votes
3answers
148 views

How many values does the expression $1 \pm 2 \pm 3 \pm \cdots \pm n$ take?

How many different values does the expression $1 \pm 2 \pm 3 \pm \cdots \pm n$ take? I was wondering about this problem and didn't think it was immediately obvious. The answer can't be $2^{n-1}$ ...
0
votes
1answer
21 views

How many n-digit numbers with strictly increasing digits do exist?$(n<10)$

How many n-digit numbers with strictly increasing digits do exist?$(n<10)$ We mean numbers like: $13458$,these numbers do not have $0$ as a digit.How can we count them?? I used trees to distinguish ...
1
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0answers
24 views

Success runs in dependent trials

There are 260 business days in a year. We have 54 employees. Each employee is required to bring donuts twice a year on different days. Each employee chooses the two days at random, independently of ...
0
votes
1answer
21 views

Drawing exactly $r$ red, $g$ yellow and $b$ blue balls out of an urn

In an urn, let there be $U \in \mathbb{N}$ balls. Of these balls, $R$ are red, $G$ are yellow and $B$ are blue, and there are no other colors than these in the urn. (So, $R + G + B = U$.) Now, without ...
0
votes
1answer
21 views

permutations of n objects

Does the number of permutations of $n$ objects, $r$ alike of one kind and $n−r$ alike of another kind, always equal the combinations of n different objects taken $r$ at a time? Explain. I know ...
2
votes
1answer
28 views

Enumerating the primitive recursive functions without repetition

According to this paper (and this one), it is possible to enumerate the primitive recursive functions without duplication, even though equality of primitive recursive functions is not decidable. I am ...
0
votes
2answers
93 views

Catalan numbers and triangulation

Assume $C_n$ is the number of triangulations of a polygon with $n+2$ sides. Using a combinatorial proof, show that $(4n+2)C_n=(n+2)C_{n+1}$. I don't even know where to start with this one. I ...
0
votes
0answers
24 views

Measure of Connectivity on a Chessboard

I'm programming a boardgame...game. The basic idea of it is there are two players (call them $X$ and $Y$) that are trying to trying to build a wall connecting the North and South, and East and West ...
0
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0answers
28 views

Multivariable recurrence: Solving $c(n,k) = c(n-1,k) + c(n-1,k-1) = \binom{n}{k}$ by algebraic methods.

Let $(c_{n,k})_{n,k=0}^{\infty}$ be defined by $c(0,0)=1$, $\:\:c(0,k)=0 \:\: \forall \: k > 0$ $$c(n,k) = c(n-1,k) + c(n-1,k-1) \:\:\: \forall \: n \geq 1$$ I can show that the ...
0
votes
2answers
29 views

How to count the number of x in a rows in a larger set.

For example, I have 4 in a row like so: xxxx I can see that it has 2 xxx in it and 3 xx. ...
1
vote
1answer
29 views

Proof by strong induction combinatorics problem

$1(1!) + 2(2!) + 3(3!) + \dots + n(n!) = (n+1)! - 1$ How do we prove this by strong induction? I know how to do it with weak induction, but how would strong induction work with this problem?
5
votes
2answers
159 views

How many integers from 43523 to 93107 contain at least one digit 7

How many integers from $43523$ to $93107$ contain the digit $7$ at least once? I know that if we had $43000$ to $93000$, we would subtract integers that do not contain digit $7$ from the total ...
5
votes
3answers
39 views

Game is winnable if and only if $n \neq k$

Integers $n$ and $k$ are given, with $n \ge k \ge 2$. You play the following game against an evil wizard. The wizard has $2n$ cards; for each $i = 1, \ldots, n$, there are two cards labelled $i$. ...
0
votes
2answers
34 views

Find $n$ such that the congruence $x^n\equiv 2\mod 13$ has a solution for $x$.

Find $n$ such that the congruence $x^n\equiv 2\mod 13$ has a solution for $x$. I am not getting any idea how to start this problem. Please give some hits
0
votes
1answer
15 views

Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given ...
2
votes
0answers
35 views

$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$ integer for $k \in \mathbb{N}$

How do I see that for any positive integer $k$,$$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$$is an integer?
0
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0answers
31 views

proof of number of sub arrays of an array of size $N$ using combinatorics

What is the proof that number of sub arrays of an array of size $N$ is $$\frac{N(N+1)}{2}$$
7
votes
2answers
532 views

Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...
0
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1answer
26 views

Literature on generating functions for networks

Are you aware of any material the presents all (or most, or many) the properties and applications of generating functions in the context of graphs? For example I am aware of 'Generating ...
0
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0answers
17 views

Can you identify this stochastic process?

So I run into this problem the other day and I cannot even think of the keywords I need to use to look it up. For the discrete random variable $X$ we have: $P_{\Delta X(t)} = F\big(X(t-1), ...
-1
votes
1answer
36 views

How do I find all n values for which the equation $\phi (n) = 8$ holds? [duplicate]

I've heard all kinds of different ways to solve this problem, yet haven't been able to apply them specifically to the number 8 (Worked fine for 6 for example). I'd love to see a well-explained ...
6
votes
1answer
109 views
+100

Counting the number of rank $r$ binary $n \times k$ matrices that has unique columns

I'm trying to figure out how many ways there are to construct a $k \times n$ binary matrix such that it has rank $r$ and no column is repeating. I've tried a bunch of different approaches. The attempt ...
3
votes
1answer
29 views

How many straight lines can be made between 10 points such that 4 of them are colinear?

So i know how to get the answer. We just have to find $C(10,2)$ and subtract $C(4,2)$ and add 1. We are basically counting all the points between co-linear points as 1. So the question is why we are ...
1
vote
1answer
25 views

How many different arrangements of triangles that are either red or blue around a regular heptagon are possible?

I have the following problem I have an yellow heptagon (regular $7$ sided polygon) Against every side there is a triangle. The triangle is either red or blue. How many different arrangements of ...
1
vote
3answers
6k views

soccer betting combinations for accumulators

I would like to bet on soccer games but I would like to place a bet on every combination possible. For example, I bet on $10$ different games, and each soccer match can go three ways: either a win, ...
1
vote
0answers
25 views

Number of ways to get from a point to another one in the plane

I was trying to solve the following problem related to "counting cases": Consider the point $(0,0)$ in the plane and another point $(m,n)$ with $m,n>0$ integers. Suppose you want to get from the ...