Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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Show that $r_k^n/n \le \binom{kn}{n} < r_k^n$ where $r_k = \dfrac{k^k}{(k-1)^{k-1}}$

Show that for $n \ge 2$, $\dfrac{r_k^n}{n+1} \le \binom{kn}{n} < r_k^n$ where $r_k = \frac{k^k}{(k-1)^{k-1}}$. This is a generalization of How to prove through induction which asks for a proof ...
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3answers
21 views

Application of the Binomial Theorm-remainder

I am having a confusion in this question- What is the remainder when $7^{103}$ is divided by 24? I attempted it as follows - It can be written as $(7^2)^{51} *7$ Which can be written as ...
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0answers
29 views

How to compute linear recurrence of a sum of binomial-multiplied linear recurrences [duplicate]

I have $$g(n) = \sum_{k=1}^{n} \binom{n}{k}f(k)$$ where $f(k)$ is a large linear recurrence. $g(n)$ is also a linear recurrence as well. Normally, when computing the value of a linear recurrence, I ...
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2answers
41 views

Bionomial therom, what's the cofficent? [on hold]

What's the cofficent of $x^3 y^3$ in $(5x+2y-3)^8$
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1answer
46 views

Simplying linear recurrence sum with binomials

Is there a way to simplify $$\sum_{k=1}^{n} \binom{n}{k}f(k)$$ Where $f(k)$ is a large linear recurrence?
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0answers
26 views

Binomials problem [on hold]

If [pi] denotes product of all binomial coefficients in income sales per fiscal year in (1 + x)^n, then the ratio of [pi]2002 to [pi]2001 can be expressed as (2002^m)/n! Therefore, find the minimum ...
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0answers
18 views

Coefficients for the falling factorial

Hello fellow mathematicians, I am trying to find a generating function, or at least find some useful property from the coefficients of the falling factorial. Let $(x)_n$ denote a falling factorial, ...
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4answers
234 views

Orthogonality for Binomial Coefficients

Could somebody explain to me where these two formulas come from as applications of the binomial theorem? $$\sum_{k=0}^n {n \choose k}(-1)^kk^r=0$$ for non-negative integers $r\lt n$. And ...
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0answers
32 views

Weighted sum of binomial coefficients by powers of lower value.

I am trying to calculate $\sum_{i=0}^n i^K {n \choose i}$ for $K \in \mathbb{N}$. Clearly, the case $K=0$ is trivially $2^n$ by the binomial theorem. For higher $K$ I am stumped. I know I can use: ...
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22 views

Binominial Theorem proving

As I was trying to understand the proof of Binomial Theorem by induction, I got stuck at this line. What formulas should be used to get from left to right part? Any explanations and answers ...
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1answer
19 views

Cofficient of x in a product

How can I efficiently find the coefficient of $x^m$ in the following product - $\prod\limits_{i=1}^{n-1}(1 - p_i + p_ix)$
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4answers
110 views

Show that $\binom{2n}{n}$ is an even number, for positive integers $n$.

I would appreciate if somebody could help me with the following problem Show by a combinatorial proof that $$\dbinom{2n}{n}$$ is an even number, where $n$ is a positive integer. I ...
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1answer
38 views

Sum of Some Binomial Terms Equals Zero

Let $q$ be a positive integer. Then the sum $$ \sum_{k=q}^\ell (-1)^{k+q}\binom{k}{q}\binom{\ell}{k} = \left\{\begin{array}{ccc} 0 \mbox{ if } \ell =q\\ 1 \mbox{ if }\ell >q\end{array}\right. ...
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1answer
35 views

Bound on the Beta function

For positive integers x and y, we have that $$ B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} = \frac{1}{x} \left( \begin{array}{c} x+y-1 \\ x \end{array} \right)^{-1} . $$ However, $$ \left( ...
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2answers
55 views

Summation over a product of binomial coefficients

Question: I can't figure out why the following equality is true $\sum_\limits{k=a-b-c}^{d} (-1)^k \binom{d}{k}\binom{k+b+c}{a} = (-1)^d \binom{b+c}{a-d} $ How can this be shown? (In the book it just ...
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0answers
17 views

Stirling or better binomial approximation

Is there a method to find a approximation to $\log_2 \dbinom{n}{n^a}$ with $a\in(0,1)$? Similar to Approximating the logarithm of the binomial coefficient however here argument scales as radical of ...
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1answer
31 views

Show that $\binom{n}{k}=\frac{1}{2i\pi}\int_{C}\frac{(1+z)^{n}}{z^{k+1}}dz.$

I would like to prove that $$\binom{n}{k}=\frac{1}{2i\pi}\int_{C}\frac{(1+z)^{n}}{z^{k+1}}dz.$$ C is the circle at $0$ with radius $r>0$. I cannot get that expression, if I write the integral as ...
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1answer
79 views

Why General Leibniz rule and Newton's Binomial are so similar?

The binomial expansion: $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$ The General Leibniz rule (used as a generalization of the product rule for derivatives): $$(fg)^{(n)} = ...
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3answers
54 views

Combinatorial proof of $ \sum \limits_{i = 0} ^{m} 2^{n-i} {n \choose i}{m \choose i} = \sum\limits_{i=0}^m {n + m - i \choose m} {n \choose i} $

I've been wondering for a while how to solve (prove) a combinatorial identity, using just combinatorial interpretation: $$ \sum \limits_{i = 0} ^{m} 2^{n-i} {n \choose i}{m \choose i} = ...
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1answer
37 views

How can find this two sequence recursive relations?

Let $$D_{n}=\sum_{j=0}^{n-1}(-1)^{n+j-1}\dfrac{\binom{2n-4}{j}}{n+j-1},E_{n}=\sum_{j=0}^{n-1}(-1)^{n+j-1}\dfrac{\binom{2n-4}{j}}{n+j}$$ I want find $D_{n}$ and $E_{n}$ recursive relations, I ...
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2answers
23 views

Is there limitation when applying binomial theorem?

Problem as title showed. $(a+b)^{-n}$. If $n$ is a positive integer. Can $a$ or $b$ be a complex number? Many thanks in advance.
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3answers
54 views

Evaluating $\sum_{0\leq k,l \leq n}\binom{n}{k}\binom{k}{l}l(k-l)(n-k)$ algebraically

I'm having problems with the following sum: $$\sum_{0\leq k,l \leq n}\binom{n}{k}\binom{k}{l}l(k-l)(n-k)$$ It's quite easy to think about it combinatorically: We have $n$ balls, we're coloring $k$ ...
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2answers
24 views

A sum expressed by a Kampe de Feriet function.

Let $a_1$,$a_2$, $a_3$ and $b_1$,$b_2$, $b_3$ be real numbers subject to $1+b_1+b_2 - b_3 > 0 $. By generalizing the result from A sum involving a ratio of two binomial factors. we have shown that ...
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45 views

Second question about a limit.

Is the following sequence converge? $$ \lim_{n\rightarrow\infty}\frac{1}{(1+M)^{2n}}\sum_{i=0}^{n}\left( \begin{array}{c} 2n ...
4
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2answers
39 views

What is wrong with ${13 \choose 1}{4 \choose 2} \cdot {12 \choose 1}{4 \choose 2}$ as combinations for two pair in poker?

Let's consider two pairs in a 52 cards deck of poker where every person gets five cards. My idea to approach this problem is to take following steps: First pair There are ${4 \choose 2}$ ...
6
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2answers
60 views

Binomal theorem show that $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\dots=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\dots=2^{n-1}$

I'm having some trouble with this question Show that $$\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\dots=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\dots=2^{n-1}$$ Attempt: Expanding $(1+1)^n=2^n$ ...
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25 views

Truncated Binomial Series

Can the truncated binomial series be expressed as a closed form \begin{align} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{k} x^{k} \end{align}
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43 views

Use induction on n for the Binomial Coefficient

Use induction on n to show that the divide and conquer algorithm for the Binomial Coefficient problem computes 2*C(n,k) -1 terms to determine C(n,k). The C(n,k) means "n choose k" I started through ...
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3answers
28 views

What is the alternating sum of coefficients and what does it have to do with the zeroes of the function?

So, my teacher told us today, while we were solving this integral: $$\int\frac{dx}{x(2x^3+x^2+1)}$$ that the alternating sum of coefficients of $2x^3+x^2+1$ is 0 (2-1+0-1=0) and hence, one zero of the ...
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1answer
77 views

What is the sum of this infinite series

So the question is - $\displaystyle S = \sum_{n=1}^\infty{\frac{1}{10^n}\left(\begin{matrix}2n\\ n\end{matrix}\right)}$. Find $S$. I tried converting the $n^{th}$ term as a difference of two terms ...
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4answers
67 views

Equality of sums

How does one show that $$\sum_{j=k}^n\binom{j-1}{k-1}q^{j-k}=\sum_{j=k}^n\binom{n}{j}p^{j-k}q^{n-j},$$ where $p+q=1$? I suppose one needs to substitute $p=1-q$ on the right side and then use the ...
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1answer
54 views

Coefficient calculation problem with $x^{50}$

I get a calculation problem, when surprisingly come to this question: $f(x)= \frac {1}{(1+x)(1+x^2)(1+x^4)} $ and try to find the coefficient of $x^{50}$ in $(f(x))^3$ sentence? how this will ...
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4answers
109 views

What is the story behind ${n+1 \choose k} = {n \choose k} + {n \choose k-1}$? [duplicate]

By exploring the inductive proof from this question I came to the point where I did not understand this step: $${n+1 \choose k} = {n \choose k} + {n \choose k-1}$$ There is a wikipedia article but ...
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1answer
37 views

Discrete Math Identity Proof Binomial Coefficients

The question is to prove this identity: ! where k, m, n ∈ Z+. Using pascal's identity on the left, so far I have: ! If m is even then they cancel each other and should equal 0. If m is odd then ...
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1answer
18 views

Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime.

Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime. Here $v_p(k)$ denotes the largest $\alpha\in\mathbb Z_{\ge 0}$ s.t. $p^\alpha\mid k$. We have ...
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2answers
88 views

Why does $\binom{n}{2} = \frac{n^2 - n }{2}$?

In a proof in Introduction to Algorithms, the book says $\binom{n}{2} \cdot \frac{1}{n^{2}} = \frac{n^2 - n }{2}\cdot \frac{1}{n^{2}}$, which implies $\binom{n}{2} = \frac{n^2 - n }{2}$. Why are ...
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1answer
62 views

How to simplify $\sum_{i=0}^{2n - d - 1} {n \choose i}$?

Is it possible and how could I simplify this sum into a formula who's quantity of operations is independent of n? $$ \sum_{i=0}^{2n - d - 1} {n \choose i} $$ ...
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1answer
38 views

Comparing Coefficients in Summations

Suppose I have the following equality: $$\sum_{k=0}^{n-a}\sum_{j=0}^{k}\binom{n}{k}\binom{k}{j}\frac{f(a,k)\cdot g(b,n-k)}{n!}=\sum_{k=0}^{n-a}\binom{n}{k}\binom{n-k}{a}\frac{z^k \cdot ...
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1answer
42 views

Binomial Coefficients

I want to get ahead in my classes and learn Binomial Theorem ahead of time. What I know so far is that the formula below is the Binomial Coefficient: $\binom n k = \frac {n!} {(n-k)!k!}$ and ...
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1answer
50 views

Calculate the sum $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$.

I want to calculate this sum: $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$. I tried to use some differentation techniques, but they didn't work. Could you help me with this?
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2answers
70 views

Proving if it is prime

I'm quite lost on how to prove things, with the $n \choose k$ and proving. So the question is: Prove that $n \choose k$ is divisible by $n$ if $n$ is a prime number and $1 \le k\le n-1$ Like, how ...
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43 views

Recursion Formula Euler Numbers

I am trying to derive the formula $$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}E_k = \displaystyle\sum_{k=0}^{n}{n\choose k}^2E_k=0$$ Where $E_k$ are the Euler Numbers. The approach that I have taken ...
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35 views

A sum involving a ratio of two binomial factors.

Let $a\ge 0$, $a_1\ge 0$ ,$b \ge 0$ and $b_1\ge0$ be real numbers subject to $1+b+a_1-b_1-a >0$. Let $m$ be a positive integer. Then using methods similar to those in Another sum involving binomial ...
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1answer
18 views

A multivariate sum that yields a closed form expression

Let $d\ge 2$ be a an integer. Let $b_1,b_2,\cdots,b_d$ be positive integers. As a by product of certain calculations I have discovered that: \begin{equation} \sum\limits_{q_2=0}^{b_2} \cdots ...
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1answer
33 views

Calculate wining probability in a dart game

Suppose we're playing the following dart game: The player can play up to $T$ rounds. In each round of the game, the player first throw a black dart, and then a white dart. Each dart independently hit ...
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463 views

Find the Pascal's Limit [closed]

Let $P_{n}$ be the product of the numbers in row of Pascal's Triangle. Then evaluate $$ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}}$$
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1answer
36 views

Find the sum of coefficient of all the integral power of $x$ in the expansion of $\big(1 + 2\sqrt x\big)^{40}$? [closed]

While going through certain question online. This question took a lot of my time. Can anyone please help me with this question!!
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0answers
30 views

Is it true that $n \choose 3$ is a perfect square for only finitely many positive integers $n$ ? [duplicate]

Is it true that the only positive integer $n$ such that $n \choose 3$ is a perfect square is only $n=50 $ ? Or can some one just tell whether is it true that $n \choose 3$ is a perfect square for only ...
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1answer
32 views

Complex analysis ~ Binomial theorem

Given the identity $ \binom {2n} {n} = \frac{1}{2\pi i} \int_{C_r} \frac{(1+z)^{2n}}{z^{n+1}}dz,$ with $C_r$ the unit circle, prove that $\forall n \in \mathbb{N}$: $\binom {2n} {n} \leq 4 ...
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1answer
19 views

prove $\mathscr P_k (A)$ has ${n \choose k} $ elements

I am working on this question If $A$ is a set and $k\in \mathbb {N}$, let $mathscr P_k$ be the set of all subsets of $A $ that have k elements. Prove $\mathscr P_k (A)$ has ${n \choose k} $ ...