  wanaopatays 2022-05-24 Answered

### Prove combinatorially the recurrence ${p}_{n}\left(k\right)={p}_{n}\left(k-n\right)+{p}_{n-1}\left(k-1\right)$ for all $0Recall that ${p}_{n}\left(k\right)$ counts the number of partitions of k into exactly n positive parts (or, alternatively, into any number of parts the largest of which has size n). osmane5e 2022-05-24 Answered

### Bounding $A\left(n,d\right)=max\left\{M|$ exists a code with parameters n,M,d}I would like to prove that this lower bound of $A\left(n,d\right)=max\left\{M|$ exists a code with parameters n,M,d} (where n is the length of the block code, M the number of words of the code, and d, the minimal distance of the code), holds:$\frac{{2}^{n}}{\sum _{i=0}^{d-1}\left(\genfrac{}{}{0}{}{n}{i}\right)}\le A\left(n,d\right)$For trying to see this, I have tried to connect this inequality with the cardinal of the ball of radius $d-1$, that is $\sum _{i=0}^{d-1}\left(\genfrac{}{}{0}{}{n}{i}\right){2}^{i}$, so for sure, that quantity is less than ${2}^{n}\sum _{i=0}^{d-1}\left(\genfrac{}{}{0}{}{n}{i}\right)$. But I don't see if this is or not helping me at all... I would appreciate some guidance, help, hint,... Thanks! il2k3s2u7 2022-05-23 Answered

### In how many ways can we distribute 2 types of gifts?The problem: In how many ways can we distribute 2 types of gifts, m of the first kind and n of the second to k kids, if there can be kids with no gifts?From the stars and bars method i know that you can distribute m objects to k boxes in $\left(\genfrac{}{}{0}{}{m+k-1}{k-1}\right)$ ways. So in my case i can distribute m gifts to k kids in $\left(\genfrac{}{}{0}{}{m+k-1}{k-1}\right)$ ways, same for n gifts i can distribute them in $\left(\genfrac{}{}{0}{}{n+k-1}{k-1}\right)$ ways. So now if we have to distribute m and n gifts we can first distribute m gifts in $\left(\genfrac{}{}{0}{}{m+k-1}{k-1}\right)$ ways, then n gifts in $\left(\genfrac{}{}{0}{}{n+k-1}{k-1}\right)$ ways, so in total we have:$\left(\genfrac{}{}{0}{}{m+k-1}{k-1}\right)\cdot \left(\genfrac{}{}{0}{}{n+k-1}{k-1}\right)\phantom{\rule{1em}{0ex}}\text{ways.}$.Is my reasoning correct?What about when we have to give at least 1 gift to each kid, can we do that in$\left(\genfrac{}{}{0}{}{m-1}{k-1}\right)\cdot \left(\genfrac{}{}{0}{}{n+k-1}{k-1}\right)+\left(\genfrac{}{}{0}{}{n-1}{k-1}\right)\cdot \left(\genfrac{}{}{0}{}{m+k-1}{k-1}\right)\phantom{\rule{1em}{0ex}}\text{ways?}$ Landyn Jimenez 2022-05-23 Answered

### How to approach this discrete graph question about Trees.A tree contains exactly one vertex of degree d, for each $d\in \left\{3,9,10,11,12\right\}$.Every other vertex has degrees 1 and 2. How many vertices have degree 1?I've only tried manually drawing this tree and trying to figure it out that way, however this makes the drawing far too big to complete , I'm sure there are more efficient methods of finding the solution.Could someone please point me in the right direction! hushjelpw4 2022-05-23 Answered

### Representing a sentence with quantified statementsMy approach to this question: $\mathrm{\exists }x\left(P\left(x\right)\to R\left(x\right)\right)$I cannot verify if my answer is correct, any help to verify my answer would be appreciated and if I did wrong any help to explain why would also be appreciated. groupweird40 2022-05-23 Answered

### Exercise involving DFTThe fourier matrix is a transformation matrix where each component is defined as ${F}_{ab}={\omega }^{ab}$ where $\omega ={e}^{2\pi i/n}$. The indices of the matrix range from 0 to $n-1$ (i.e. $a,b\in \left\{0,...,n-1\right\}$)As such we can write the Fourier transform of a complex vector v as $\stackrel{^}{v}=Fv$, which means that${\stackrel{^}{v}}_{f}=\sum _{a\in \left\{0,...,n-1\right\}}{\omega }^{af}{v}_{a}$Assume that n is a power of 2. I need to prove that for all odd $c\in \left\{0,...,n-1\right\}$, every $d\in \left\{0,...,n-1\right\}$ and every complex vectors v, if ${w}_{b}={v}_{cb+d}$, then for all $f\in \left\{0,...,n-1\right\}$ it is the case that:${\stackrel{^}{w}}_{cf}={\omega }^{-fd}\phantom{a}{\stackrel{^}{v}}_{f}$I was able to prove it for $n=2$ and $n=4$, so I tried an inductive approach. This doesn't seem to be the best way to go and I am stuck at the inductive step and I don't think I can go any further which indicates that this isn't the right approach.Note that I am not looking for a full solution, just looking for a hint. cyfwelestoi 2022-05-23 Answered

### Combinatorial proof of ${P}_{r}^{n+1}=r!+r\left({P}_{r-1}^{n}+{P}_{r-1}^{n-1}+\cdots +{P}_{r-1}^{r}\right)$, where ${P}_{r}^{n}$ denotes the number of r- permutations of an n element set. realburitv4 2022-05-23 Answered

### How to solve $⌊x⌋+⌊\frac{1}{x}⌋=1$? Richardtb 2022-05-23 Answered

### Solution of linear congruence systemSolve Now I have read about the Chinese Remainder Theorem, which is particularily helpful in solving systems of linear congruence, but in this notice that the moduli are not relatively prime (pairwise). So, I cannot apply the theorem. In fact this leads me to a general question: If we are given as system:${a}_{1}x\equiv {b}_{1}\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}{n}_{1}\right)\phantom{\rule{0ex}{0ex}}{a}_{2}x\equiv {b}_{2}\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}{n}_{2}\right)\phantom{\rule{0ex}{0ex}}{a}_{3}x\equiv {b}_{3}\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}{n}_{3}\right)\phantom{\rule{0ex}{0ex}}{a}_{4}x\equiv {b}_{4}\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}{n}_{4}\right)\phantom{\rule{0ex}{0ex}}...\phantom{\rule{0ex}{0ex}}{a}_{r}x\equiv {b}_{r}\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}{n}_{r}\right)$where the moduli are not pairwise relatively prime. How do we go about solving this system and what are the conditions that determine the solvability of the system. Emery Boone 2022-05-22 Answered

### You have {a,b,c} as your character set. You need to create words having 10 characters that must be chosen from the character set. Out of all the possible arrangements, how many words have exactly 6 as? qtbabe9876a9 2022-05-22 Answered

### How to expand the following notation?Can someone help me to expand the below equation for $N=2$ and $N=3$?${f}_{i}=\sum \left\{{e}_{i}:j\le i\le k\right\}$ for some $1\le j\le k\le N$.Here ${e}_{i}$ is the unit vector of coordinate i in an N-dimensional space. Alani Conner 2022-05-22 Answered

### The following cards are dealt with 3 people at random so that every one of them gets the same number of cards:${R}_{1},{R}_{2},{R}_{3},{B}_{1},{B}_{2},{B}_{3},{Y}_{1},{Y}_{2},{Y}_{3}$where R, B and Y denote red, blue and yellow, respectively. Find the probability that everyone gets a red card. Richardtb 2022-05-22 Answered

### How to inductively prove a graph property?I'm stuck on Part 1, I can't find it inductively. The distance from $\left[k,k+1\right]$ is going to be a non-zero positive value with a vertex. Therefore, there is going to be a positive edge weight from w(uv) that goes from $\left[k,k+1\right].$. Therefore, there exists a v on the set of V such at $A\left[u,k+1\right]=A\left[v,k\right]+w\left(uv\right).$.The ProblemGiven an undirected graph $G=\left(V,E\right)$ with positive edge weights w(e) for each edge $e\in E$, we want to find a dynamic programming algorithm to compute the longest path in G from a given source s that contains at most n edges.To do this first define A[v,k] as the weight for the longest path from node s to node v of at most k edges.1. First we need to prove an optimal sub-structure by induction. Show that if A[v,k] is the weight of the longest path, then for all $u\in V$, there exists a $v\in V$ such that $A\left[u,k+1\right]=A\left[v,k\right]+w\left(uv\right)$.2. Describe a dynamic programming algorithm that finds the optimal length using part 1. Specifically: describe (1) the OPT recurrence (2) the running time of the iterative solution for computing the OPT table. Aidyn Cox 2022-05-22 Answered

### A relation that is both an equivalence relation and a function.The question is: "Suppose R on A is both an equivalence relation on A and a function from A to A. What is R?" I'm not entirely sure where to start, I'm not even sure I understand what a function from A to A represents, can anybody help with this? Hailey Newton 2022-05-22 Answered

### How many odd 6 digit numbers can be formed by using the digits 1,2,3,4,5,6 which are divisible by 3?I have found that we have $3\cdot {6}^{5}$ odd numbers. I understand, that if number is divisible by 3, the sum of its digits also should be divisible by 3, but I don't know how to use this... skottyrottenmf 2022-05-21 Answered

### Proving two integers are relatively prime using Bezout's Theorem.If $\mathrm{gcd}\left(a,b\right)=1,$, a and b are relatively prime and also if gcd(a,b) is equal to 1 there should be 2 integers k and m such that $ak+bm=1.$.If we can find such integers k and m, is it a proof that a and b are relatively prime ?What you think about that proof? Is it correct way? Marianna Stone 2022-05-21 Answered

### How to prove $A+{A}^{\prime }B+{A}^{\prime }{B}^{\prime }C+{A}^{\prime }{B}^{\prime }{C}^{\prime }D=A+B+C+D$Prove the above relationship by using the Boolean definition. I tried $A+{A}^{\prime }B=A+B$, but end up with $A+B+{A}^{\prime }{B}^{\prime }\left(C+D\right)$, how can I go next? Liberty Mack 2022-05-21 Answered

### Does LCM have distributive property of multiplication?Is statement $lcm\left(ax,ay\right)=a\ast lcm\left(x,y\right)$ true? (assuming all variables are positive integers) Brooke Webb 2022-05-21 Answered

### Why does "Some student has asked every faculty member a question" translate to $\mathrm{\forall }y\left(F\left(y\right)\to \mathrm{\exists }x\left(S\left(x\right)\vee A\left(x,y\right)\right)\right)$ ?Context: I'm in undergrad discrete math, this is a textbook question from Discrete Mathematics and its Applications 7th editionHere's the question:Let S(x) be the predicate "x is a student," F(x) the predicate "x is a faculty member," and A(x, y) the predicate "x has asked y a question," where the domain consists of all people associated with your school. Use quantifiers to express the following statement.Some student has asked every faculty member a question.This is what the textbook says is the correct answer:$\mathrm{\forall }y\left(F\left(y\right)\to \mathrm{\exists }x\left(S\left(x\right)\vee A\left(x,y\right)\right)\right)$I mostly understand how this is correct, but shouldn't it be $\wedge$ instead of $\vee$? hughy46u 2022-05-21 Answered

### Bounding the order of tournaments without transitive subtournaments of certain size.A tournament of order N is a directed graph on [N] obtained by assigning a direction to each edge of ${K}_{N}$. A tournament D is transitive if for every triple $a,b,c\in N$, $\left(a,b\right),\left(b,c\right)\in E\left(D\right)$ implies that $\left(a,c\right)\in E\left(D\right)$. For $n\in \mathbb{N}$ let f(n) be the maximum integer such that there exists a tournament of order f(n) without a transitive sub-tournament of size n. Show that $f\left(n\right)>\left(1+o\left(1\right)\right){2}^{\frac{n-1}{2}}$.

Students pursuing advanced Math are constantly dealing with advanced Math equations that are mostly used in space engineering, programming, and construction of AI-based solutions that we can see daily as we are turning to automation that helps us to find the answers to our challenges. If it sounds overly complex with subjects like exponential growth and decay, don’t let advanced math problems frighten you because these must be approached through the lens of advanced Math questions and answers. Regardless if you are dealing with simple equations or more complex ones, just break things down into several chunks as it will help you to find the answers.