combinatorics - Help me proving $\sum_{k=0 n {n \choose k}k^r x^k = \sum_{j=0

I have trouble proving the polynomials are identity: $\sum_{k=0}^n {n \choose k}k^r x^k = \sum_{j=0}^{r} {n \choose j} j! (1+x)^{n-j} x^j S(r,j)$ $S(r,j)$ is a Stirling number of the second kind.

real analysis - I need some help, with a dense set S in $\mathbb{R}$ and $s\in S

I need to prove that if $\emptyset \neq S \subset \mathbb{R} \text{ and } s \in S, \text{ then } S \backslash \{s\}$ is also dense in $\mathbb{R}$. Would this result imply that if $\emptyset \neq S \

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Help in solving a question on Planarity graphs Given that r is even show that t(

Given that r is even, (I) show that t(Kr,s) = r/2. (II) if r is even and s > (r - 2)^2 then .t(Kr,s) =r/2.

R O U T I N E S (@morning..routines..ideas)

TikTok (틱톡) 의 R O U T I N E S (@morning..routines..ideas) |R O U T I N E S If you need help for your routine message me and I will help you.R O U T I N E S (@morning..routines..ideas) 님의 최신 동영상을 시청하세요.

abstract algebra - Show ring isomorphism $\mathrm{End R ( \bigoplus i=1}^r S_i

$\newcommand{\End}{\mathrm{End}}$ I have been struggling with this Representation Theory question for the past week: Let $R$ be a ring and $S_1, \ldots, S_r$ simple $R$-modules with $S_i$ not isomo...

probability theory - $S_n$ is distributed according to $\sim \max\{V_i : 1\leq i

If we have a random variable $S_n \sim R_1+\cdots +R_n$ where $R_j\sim \operatorname{Exp}(j\lambda)$ and independent, is it true that also $S_n \sim \max\{V_i : 1\leq i \leq n\}$ with $V_i \overset...

algebra precalculus - Prove that if $r,s,t \in \Bbb{N}$, then $2^r + 2^s = 2^t \

I'm trying to prove the following statement: If $\{r,s,t\} \subset\Bbb{N}$ then $$2^r + 2^s = 2^t \iff r=s.$$ (We assume $0 \notin \Bbb{N}$ for this problem) Here is how I attempted to prove ...

real analysis - Help With Basic Proof from Rudin PMA Chapter One - Mathematics S

From Walter Rudin's Principles of Mathematical Analysis, Third Edition, page 20, Step 8: I want to complete the proof of (b) by proving the following claim. Claim: For $r\in{}\mathbb{Q}^+$ and $s\...

discrete mathematics - Need help to prove $(R\circ S \circ R)^n \subseteq (R \ci

There is a relation $R$ and $S$ on set $U$. Given $R$ is transitive. I need to prove $(R\circ S \circ R)^n \subseteq (R \circ S)^n \circ R$ for all $n \geq 1 $ My Attempt:- Tried doing this by