2
votes
1answer
20 views

number of ways to put 4 black,4 white,4 red balls in 6 different boxes

The question says:in how many ways we could put 4 black,4 white,4 red balls in 6 different boxes? boxes are distinguishable,black balls are identical,red balls are identical,and white balls are ...
0
votes
0answers
59 views

Balls and Boxes [closed]

How many ways are there to put 6 balls in 3 boxes if: a)the balls are not distinguishable and neither are the boxes? b)the balls are not distinguishable but the boxes are? c)the balls are ...
0
votes
1answer
30 views

Permutation problem - How many permutations are there such that no two numbers are immediately adjecent? [duplicate]

Consider the set of numbers 1,1,2,2,3,3,4,4. How many permutations are there such that no two identical numbers are immediately adjacent?
1
vote
3answers
52 views

Question regarding permutations and combinations?

Hi, I was just wondering on how you are supposed to approach this question. I keep getting 114 as an answer, but the answers say it is 174. How would anyone do this question, because I feel like I'm ...
0
votes
0answers
57 views

Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
4
votes
4answers
399 views

Probablity that 3 husbands sit next to their wives round a circular table

There are 3 couples sitting randomly round a 6-seater circular table. What is the probability that all the husbands and wives sit next to each other? My attempt: First wife, say, takes any of the ...
1
vote
2answers
56 views

Semi-Ambiguous Combinatorics Problem

I'm having trouble fully understanding this introductory combinatorics problem; this is all that's provided, so it's most likely me not fully comprehending the question. Any help is appreciated. The ...
0
votes
0answers
34 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
0
votes
1answer
31 views

Selecting 6 people from a group of 10 people with special conditions

Sorry for a misleading or such title, but i didn't know how to make it short enough. Anyways, if we have 10 people in a group such that 8 people eat apples, 1 eats pears and one eats watermelons, ...
1
vote
2answers
38 views

Generating function satisfying a second degree equation

I got this problem in an exercise list: Let $G(x)$ be the generating function of the numeric sequence $(C_n; n \geq 0)$ satisfying the recurrence equation: $$C_n = \sum_{k=0}^{n-1}C_kC_{n-k-1}, ...
2
votes
1answer
26 views

Binomial thereom to figure out coefficents

Use the binomial theorem to find the coefficient of $x^8y^5$ in $(x + y)^{15}$ My textbook shows how to do this looking at the coefficents of Pascal's triangle but, I know theres another way using ...
2
votes
3answers
19 views

Solution check for counting in a list

This problem involves lists made from the letters T,H,E,O,R,Y, with repetition allowed. How many 4-letter lists are there that don’t begin with T, or don’t end in Y ? Just want to make sure my ...
1
vote
1answer
63 views

Divide and Conquer (recurrence relation problem)…

The problem: (a) Use a divide-and-conquer approach to devise a procedure to find the largest and next-to-largest numbers in a set of n distinct integers. (b) Give a recurrence relation for ...
0
votes
1answer
39 views

Straight in 52cards+2Joker

I have 52card (ace to king) + 2Joker I'm supposed to compute how much straights of 5 cards I can make, excluding the straight flushes (straights with all cards being the same color) My reasoning is : ...
1
vote
2answers
88 views

Combinatorial Identity involving Binomial coefficient

I’m struggling to prove this identity: $$ \sum_{k=0}^n {n \choose k} \frac{1}{k+1} = (2^{n+1}-1) \frac{1}{n+1} $$ No clue where to start but my intuition on the RHS says that there is a n+1-element ...
0
votes
1answer
122 views

simple combination

How much words in length 9 can be write by the letters: A,B,C,D,E When: The letter A appears exactly three times? We will use no more than two different letters? (Simple: AAABBABBA) No letter will ...
1
vote
3answers
42 views

Binary sequences of lenght n, with no two consecutive zeros and if starts with zero has to end with one

What is the number of binary sequences of length $n$, with no two consecutive zeros, and if starts with $0$ has to end with $1$. Would appreciate suggestions and help. I tried counting the total ...
1
vote
1answer
33 views

Number of derangements s0 $4≤f(1)$

I'm guessing it's $D_n - 2$ when $D_n = n!\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}$ Am I right?
3
votes
4answers
111 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
1
vote
2answers
54 views

A problem with my reasoning in a problem about combinations

I was given the following problem to solve: A committee of five students is to be chosen from six boys and five girls. Find the number of ways in which the committee can be chosen, if it ...
2
votes
5answers
61 views

A machine has $9$ switches. Each switch has $3$ positions. How many different settings are possible?

A machine has $9$ switches. Each switch has $3$ positions. $(1)$ How many different settings are possible? Each switch has $3$ different settings and we have $9$ total. So, $3^9=19,683$ Now, the ...
3
votes
3answers
63 views

Probability a 9-digit number has the digits 2,4, and 6 next to each other.

The integers $1,2,3,....,9$ are arraned (at random) in a row, resulting in a $9$-digit integer (without replacement). What is the probability that: The result is even? $\frac49$ or $\frac{4(8!)}{9!}$ ...
0
votes
3answers
80 views

Ice Cream Combinatorics

An ice cream store offers 14 different flavors. Customers can purchase a single scoop or a double scoop ice cream. The double scoop portion DOES NOT allow two scoops of the same flavor. How many ...
1
vote
3answers
63 views

A fair six sided die is rolled $5$ times. What is the probability that a perfect square will appear exactly twice?

I know that these rolls are independent events and order does not matter so this is a combination problem. I know that on a six sided die there are $2$ perfect squares $(1,4)$. Do I just do $5 \choose ...
0
votes
3answers
54 views

In how many ways can 9 people stand in a row for a group photograph, if 3 people are a family and need to stand together?

Are the three family members treated as one person so that the combination is: $_9C_6$ or is just $_9C_3$ (three family members). Thanks for any help.
1
vote
2answers
50 views

How many words can be formed from 'alpha'?

How many words can be formed from the word 'alpha'? The letter 'a' may be used twice but the other letters may only be used once. There are no restrictions on whether or not they're real words, just ...
4
votes
1answer
63 views

Extracting a coefficient from a generating function

Background: I am working on an exercise relating to Skolem $k-$subsets with index $k$ in Goulden and Jackson's Combinatorial Enumeration text and they broke it down to finding the coefficient of $x^n$ ...
1
vote
2answers
33 views

Number of groups that can be formed

In a tennis tournament, there are $10$ players. In the first round, $5$ groups(of 2 players) will be formed among them and elimination matches will be held among the two players in each group. In how ...
0
votes
1answer
132 views

Prove that $\lambda(v-1) = r(k-1)$

This is to do with balanced incomplete block design. Some homework exercise wants me to prove the relation $$\lambda(v-1) = r(k-1)$$ $v$ is the number of elements in your ground set. $r$ is the ...
2
votes
2answers
73 views

Number of lines determined given a set of points

Consider the following set of points in the $x-y$ plane: $$A=\{ (a,b)|a,b \in \mathbb{Z} \ and \ |a|+|b|\le 2\}$$ How to find the number of straight lines which pass through at least 2 points in ...
0
votes
2answers
43 views

Prove a formula about binoms

I want to prove that $\binom{n}{n/2} \leq 2^{n-1}$ [Assuming $n$ is even] I've tried to do that but I didn't succeed.
2
votes
1answer
44 views

Number of nodes with an even number of children in an ordered tree (AKA Plane Planted Tree)

I am looking for verification for my attempt at the solution. I have found that my answer disagrees with an answer I found here: Extract Coefficients From A Function Problem at hand: For a plane ...
2
votes
2answers
101 views

Pigeonhole principle and finite sequences

Suppose we have $75$ boxes that are labeled from $1$ to $75$ and that in each box there is at least one ball, but there are not more than $125$ balls total. I'm trying to find the largest number $n ...
-1
votes
0answers
18 views

The number of antisymmetric relations which does not contain $(a,b)$

I can not find the solution for that If $A=\{a,b,c,d\}$, determine the number of relations on $A$ that are antisymmetric and do not contain $(a,b)$. I guess the answer is $2^5\cdot3^5$ But I am not ...
10
votes
1answer
203 views

Prove that $\exp(\log(\frac{1}{1-x})) = \frac{1}{1-x}$

I am trying to prove this directly by comparing the coefficients in the two series rather than using formal calculus. Here is what I have so far, but I think I made a mistake: \begin{align*} ...
1
vote
2answers
49 views

Simple (not for me) combinatorics question

There are four balls of different colors, and four boxes of colors, same as those of the balls.What are the number of ways, in which, the balls, one each in a box, could be placed, such that a ball ...
6
votes
2answers
214 views

A problem based on pigeonhole

Numbers 1 to 1994 are divided into 6 sets.Show that at least in one group there will be two numbers whose sum is also in that group ? We can prove that at least one group will contain more than 332 ...
1
vote
1answer
40 views

Counting Urn Problem

An urn contains eight red balls labeled 1-8, seven blue balls labeled 9-15 and five yellow balls labeled 16-20. Drawing balls with replacement from the urn, what is the probability the first yellow ...
0
votes
1answer
39 views

What's the most combinations i can have in a matrix?

Imagine a chess board. I am in the midle of the board, and i can make a max of 23 jumps in a row. Houses that i was in can't be reviseted and i can only jump to the surrounding houses. How much ...
-2
votes
1answer
26 views

A cricket team of 11 players is to be formed from 20 players including 6 bowlers and 3 wicket keppers [closed]

A cricket team of 11 players is to be formed from 20 players including 6 bowlers and 3 wicket keppers.The number of ways in which a team can be formed having exactly 4 bowlers and 2 wicket keepers is
7
votes
2answers
257 views

An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$

Let $c\in\mathbb{R}\setminus\{ 1\}$, $c>0$. Let $U_i = \left\lbrace U_{i, 0}, U_{i, 1}, \dots \right\rbrace$, $U_i\in\mathbb{R}^\mathbb{N}$. We know that ...
0
votes
1answer
43 views

Does the Cayley digraph $C$= [(12)(34),(123):$A_4$] have a Hamiltonian Circuit?

This is a problem I'm working on for a friend of mine. I haven't been able to solve it, or make much progress. I have drawn the digraph, and it consists of four directed cycles of three vertices all ...
0
votes
1answer
77 views

The number ${2.2^{1000000}\choose 2^{1000000}}$ is larger than $2^{(2.2^{1000000})}$ . [closed]

The number $\left(\begin{matrix}2.2^{1000000}\\ 2^{1000000} \end{matrix}\right)$ is larger than $2^{2.2^{1000000}}$ . Can any one please help me with this question. IS THE ANSWER FALSE OR TRUE. i ...
1
vote
1answer
32 views

Set Questions? Need someone to help show

I tried doing these and I have the answers, but I can only do A. Can someone show me how to do the rest? I have the answers but don't know how to get to them..
0
votes
1answer
41 views

A committee requires one accountant, two marketing agents..

A committee requires one accountant, two marketing agents, and four board members. If there are four accountants, three marketing agents, and seven board members available for selection in the ...
0
votes
1answer
37 views

Combinatorics: help with the Inclusion-exclusion principle

$A=\{1,2,3,4\}$ $B=\{5,6,7,8,9\}$ $K$ is a relation from $A$ to $B$ ($K\subseteq AXB)$ in how many $K$'s - $1 \notin domain(K)$ ? in how many $K$'s - $\{1,2,3\}\subseteq domain(K)$ (use the ...
0
votes
1answer
32 views

Combinatorics: calcaulating number of sections of an hesse diagrem, (help with sigma).

in this picture you can see the hesse diagrem of $\subseteq$ over $P(\{x,y,z\})$ it has 12 sections. for the set $A$ with $k$ elements, $k>0$ find the numbers of sections in the hesse diagrem ...
0
votes
1answer
75 views

Combinatorics: calcaulating options of valid password of length 5 or 6 from letters and numbers

I did the following excercise using the Inclusion–exclusion principle, that's how we should do that excercise, but the answer does not match my regular calcaulation, why? The user is required to ...
4
votes
2answers
102 views

Formula for the number of integer solutions of an equation (using generating functions)

Let $a_n$ be the number of integer solutions of $$i+3j+3k=n$$ where $i \geq 0, j \geq 2, k \geq 3$. I want to use the generating function of $(a_n)_{n \in \mathbb N}$ to get a formula for $a_n$. We ...
2
votes
3answers
329 views

Number of ways to order numbers such that greatest number proceeds smallest number

With a set of numbers of length n, how many ways are there to order the set such that the greatest number always proceeds the smallest number. For example, if n = 3, there are 6 total permutations, ...