  Timiavawsw9 2022-05-21 Answered

### Find the sequence ${a}_{k}$ for generating function ${\left(\frac{1-{x}^{3}}{1-x}\right)}^{n}$.We know that $\frac{1}{\left(1-x{\right)}^{n}}=\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{k+n-1}{n-1}\right){x}^{k}$I also worked out $\left(1-{x}^{3}{\right)}^{n}$ using the binomial theorem and got $\left(1-{x}^{3}{\right)}^{n}=\sum _{i=0}^{\mathrm{\infty }}\left(-1{\right)}^{i}\left(\genfrac{}{}{0}{}{n}{n-i}\right){x}^{3i}$.I'm not sure what to do with these to get ${a}_{k}$ from $\sum _{k=0}^{\mathrm{\infty }}{a}_{k}{x}^{k}$ or if these are even what I need to solve the problem hughy46u 2022-05-21 Answered

### Probability or expected value of hirings the company will make?A company is hiring for an open position and has n interviews set up, one per day. Each day, if the candidate is better than the current employee, the employee is fired and the candidate is hired. Otherwise, the current employee keeps the job. What is the expected number of hirings the company will make?I got the answer of $\frac{1+\left(n-1\right)}{2}$. Is this correct? I used the probability of no new hires and all new hires. Laurel Yoder 2022-05-21 Answered

### The Relation between Setsif I have a relation between $A×A$if $A=\left\{1,2,3\right\}$if $B=\left\{1,2,3\right\}$$R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right)\right\}$can I say that the Relation R is Reflexive and also a Symmetric because I have (a,b) and (b,a) and also (a,a). Liberty Mack 2022-05-21 Answered

### Partitions of n where every element of the partition is different from 1 is $p\left(n\right)-p\left(n-1\right)$I am trying to prove that p(n| every element in the partition is different of $1\right)=p\left(n\right)-p\left(n-1\right)$, and I am quite lost... I have tried first giving a biyection between some sets, trying to draw an example in a Ferrers diagram and working on it... Nevertheless, I have not obtained significant results. Then, I have thought about generating functions; we know that the generating function of $\left\{p\left(n\right){\right\}}_{n\in \mathbb{N}}$ is $\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-{x}^{i}}$, so $\left\{p\left(n-1\right){\right\}}_{n\in \mathbb{N}}$ will have $\prod _{i=1}^{\mathrm{\infty }}\frac{x}{1-{x}^{i}}$ as generating function. So, what we have to prove is that $\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-{x}^{i}}-\prod _{i=1}^{\mathrm{\infty }}\frac{x}{1-{x}^{i}}=\left(1-$-x). $\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-{x}^{i}}$ is the generating function of p(n|every element in the partition is different of 1)... but i'm am not seeing why! Any help or hint will be appreciate it! Erick Clay 2022-05-21 Answered

### The random variable X has homogeneous distrigutionI have a problem with exercise:The random variable X has a homogeneous distribution in the interval [0,1]. Random variable $Y=max\left(X,1/2\right)$. Please find the expected value of a random variable Y. I need to do this for discrete variables and also for continuous variable Aiden Barry 2022-05-21 Answered

### There is a subset $T\subset S$ with $|T|=k+1$ such that for every $a,b\in T$, the number ${a}^{2}-{b}^{2}$ is divisible by 10.Let $k\ge 1$ be an integer.If S is a set of positive integers with $|S|=N$, then there is a subset $T\subset S$ with $|T|=k+1$ such that for every $a,b\in T$, the number ${a}^{2}-{b}^{2}$ is divisible by 10.What is the smallest value of N as a function of k so that the above statement is true?Step 2I have observed that perfect squares end with 0,1,4,5,6 and 9. If we have two perfect squares that end with the same as one of 0,1,4,5,6 and 9, then we are done.I think by PHP we should have $⌈\frac{N}{k}⌉=6$ Elisha Kelly 2022-05-21 Answered

### Discrete Mathematics what exactly does from A to B mean?Like if there's some relation R from a set A to a set B what exactly does this mean?I know it's just a subset of the Cartesian product $A×B,$, but what exactly does from A to B mean..? Like obviously from B to A is different so can anyone explain what the FROM really means here? Simone Werner 2022-05-21 Answered

### Proof that $⌊-x⌋=-⌈x⌉$I wanted to ask if this kind of reasoning for proving the result in the title could be considered correct:We know that: $⌈x⌉=n$ if and only if $n-1Then $-⌈x⌉=-n$ if and only if $-n-1Then multiplying by -1 the formula $-n-1 we get $n+1>-x\ge n$, inverting the sign.But $n+1>-x\ge n$ is equivalent to $n\le -x.We know that $⌊x⌋=n$ if and only if $n\le x.So from $n\le -x we can infer that $⌊-x⌋=$ $n=-⌈x⌉$ Simone Werner 2022-05-21 Answered

### Given $f:E\to E$ and $f\circ f=f$. Prove that if f is surjective or injective then $f=I{d}_{E}$. Aidyn Cox 2022-05-21 Answered

### Find all complex numbers z such as z and 2/z have both real and imaginary part integersI am really struggling to solve this one. I feel like I am missing the key part of the solution, so I would like to see how it's done.Find all complex numbers $z=x+yi$ such as z and $2{z}^{-1}$ have both real and imaginary part integersThis is what I thought:$2{z}^{-1}=\frac{2}{z}=\frac{2}{x+yi}=\frac{2}{x+yi}\cdot \frac{x-yi}{x-yi}=\frac{2x-2yi}{{x}^{2}+{y}^{2}}.$.In order to $2{z}^{-1}$ have its imaginary part $\in \mathbb{Z}$, we should equal 2y to 0$2y=0⇒y=0$$yi=0$ is an integer.x must also be an integer. We simply assume $x\in \mathbb{Z}$ (no matter what value x has, as long as it's an integer, we are good).We do the same for z and find out the same values $yi=0$ and $x\in \mathbb{Z}$.Therefore, the set is the set of all complex numbers whose real and imaginary part are integers. vamosacaminarzi 2022-05-21 Answered

### A box contains 20 green balls and 25 red balls. Two balls are chosen at random, one after the other, without replacement?i. What is the probability that both balls are red?ii. What is the probability that the second ball is red but the first ball is not?iii. What is the probability that the second ball is green? America Ware 2022-05-21 Answered

### Proving $\sum _{i=1}^{n-1}\frac{1}{lcm\left({a}_{i},{a}_{i+1}\right)}<1$ for a set of increasing positive integers.Assume ${a}_{1},{a}_{2},...,{a}_{n}$ are numbers $\in \mathbb{N}$ such that ${a}_{1}<{a}_{2}<...<{a}_{n}$ prove: $\sum _{i=1}^{n-1}\frac{1}{lcm\left({a}_{i},{a}_{i+1}\right)}<1$ Landyn Jimenez 2022-05-21 Answered

### Untyped λ-calculus: proof that for any binary relation $R\models ◊⇒{R}^{\ast }\models ◊$I'm currently in the process of reading Barendregt's "The Lambda Calculus - Its Syntax and Semantics" (1985 revised edition) and I've stumbled across a lemma whose proof I can't quite comprehend. The lemma (3.2.2) states that for all binary relations R on a set the following holds:$R\models ◊⇒{R}^{\ast }\models ◊$, where $◊$ is the diamond property of a relation:$R\models ◊\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }M,{M}_{1},{M}_{2}\left[\left(M,{M}_{1}\right)\in R\wedge \left(M,{M}_{2}\right)\in R⇒\mathrm{\exists }{M}_{3}\left[\left({M}_{1},{M}_{3}\right)\in R\wedge \left({M}_{2},{M}_{3}\right)\in R\right]\right]$, and R∗ is the transitive closure of R:$\mathrm{\forall }M,N\left[\left(M,N\right)\in R⇒\left(M,N\right)\in {R}^{\ast }\right]$ and $\mathrm{\forall }M,N,L\left[\left(M,N\right)\in {R}^{\ast }\wedge \left(N,L\right)\in {R}^{\ast }⇒\left(M,L\right)\in {R}^{\ast }\right)\right].$. Carlie Fernandez 2022-05-20 Answered

### Normal disjunctive and conjuctive form from a truth tableLet's say that we get a table with zeros and ones. We need to get it into disjunctive normal form or conjuctive normal form. We also have discrete variables ${x}_{1},..,{x}_{n}$ that are either 1 or 0. How do you determine where to put negation and where not to put it.for instance: we have a row:$p=0,q=1,r=0,\phantom{\rule{1em}{0ex}}\text{table row result = 1}$Should I write this as: $...\vee \left(\mathrm{¬}p\wedge q\wedge \mathrm{¬}r\right)\vee ...$or $...\wedge \left(\mathrm{¬}p\vee q\vee \mathrm{¬}r\right)\wedge ...$What is the correct way ? What if the table row result would be zero?Or the other way with negations? So my question is how do we know where the negations are? madridomot 2022-05-20 Answered

### How to find the probability and generating function for this problem?We toss a coin k times which is having probability p of landing on heads and a probability $q=1-p$ of landing on tails. The probability of getting exactly n heads is denoted by ${a}_{n}$. What is ${a}_{n}$ and the generating function of the sequence $\left({a}_{n}\right)$? hughy46u 2022-05-20 Answered

### A Combinatorial Problem from HMMT-2009Given a rearrangement of the numbers from 1 to n, each pair of consecutive elements a and b of the sequence can be either increasing or decreasing. How many rearrangements have of the numbers from 1 to n have exactly two increasing pairs? Meera Sunker 2022-05-20

### 1a.  Show from first principles, i.e., by using the definition of linear independence,that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrixA with associated eigenvector v = u + iw, then the two real solutionsY(t) = eat(u cos bt − wsin bt)andZ(t) = eat(u sin bt + wcos bt)are linearly independent solutions of ˙X = AX. 1b. Use (a) to solve the system (see image)  Meera Sunker 2022-05-20

### Reduce the system(D2 + 1)[x] − 2D[y] = 2t(2D − 1)[x] + (D − 2)[y] = 7.to an equivalent triangular system of the formP1(D)[y] = f1(t)P2(D)[x] + P3(D)[y] = f2(t)and solve. Despiniosnt 2022-05-20 Answered

### A particular solution for the diffirencial equation: $\left(x+i\right){y}^{\prime }+y=2x\mathrm{arctan}x$the original equation was: $\left(x+i\right){y}^{\prime }+y=1+2x\mathrm{arctan}\left(x\right)$I solved for $\left(x+i\right){y}^{\prime }+y=0$then $\left(x+i\right){y}^{\prime }+y=1$ now I'm working on the second term ($2x\mathrm{arctan}\left(x\right)$) raulgallerjv 2022-05-20 Answered

### Complement of a Set: What do I do when a set contains an element that is not in the Universal SetLet $U=\left\{x:x\in \mathbb{N},x>10\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}x<40\right\},\phantom{\rule{4pt}{0ex}}A=5,10,20,40$.The complement of the set should be${A}^{c}=\left\{\text{All natural numbers between greater than 10 and less than 40 except for 20}\right\}$ right?

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.