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

learn more… | top users | synonyms (1)

2
votes
2answers
118 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)} = ...
1
vote
3answers
89 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} = ...
1
vote
1answer
46 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 ...
0
votes
2answers
27 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.
4
votes
3answers
55 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 $k$ ...
1
vote
2answers
36 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 ...
0
votes
0answers
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
votes
2answers
58 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
votes
2answers
72 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 $(1+1)^n=2^n$ ...
1
vote
0answers
36 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}
0
votes
1answer
71 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 ...
0
votes
3answers
30 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 ...
3
votes
1answer
84 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 ...
3
votes
4answers
69 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 ...
4
votes
4answers
127 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 ...
2
votes
1answer
57 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 ...
2
votes
1answer
37 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 ...
2
votes
2answers
93 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 ...
0
votes
1answer
66 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? $$ \sum_{i=0}^{2n - d - 1} {n \choose i} $$ ...
2
votes
1answer
39 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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
52 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?
2
votes
2answers
71 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 ...
0
votes
0answers
52 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 ...
0
votes
1answer
41 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 ...
0
votes
1answer
19 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 ...
0
votes
1answer
48 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 ...
7
votes
3answers
512 views

Find the Pascal's Limit [closed]

Let $P_{n}$ be the product of the numbers in row of Pascal's Triangle. Then evaluate $$ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}}$$
0
votes
1answer
179 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!!
1
vote
1answer
37 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 ...
0
votes
1answer
24 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} $ ...
0
votes
1answer
37 views

Simplifying $\binom{n}{k}$ / $\binom{n}{k-1}$

So the question is as follows: Simplify $$ \frac{\binom{n}{k}}{\binom{n}{k-1}}. $$ And this is what I got: $$\begin{align*} \frac{\binom{n}{k}}{\binom{n}{k-1}} &= ...
1
vote
0answers
11 views

The $k$_th coefficient of the polynomial :$f_n(z)f_m(jz)+f_m(z)f_n(jz) $

Let $j=e^{2i\pi/3}$ ( $i$ is the complex number $i^2=-1$), and let : $$f_n(z)=(1+z)^n$$ Question Is there an expression (without using sums) of the $k$_th coefficient of the following polynomial ...
1
vote
3answers
37 views

Binomial Theorem Help

In my assignment I have the following question: Alan
0
votes
1answer
62 views

Sequence closed expression or others

What are closed expression or any other expression (involving integrals, specials functions...) for $\sum_{k=0}^{n}(n-2k)^t\frac{n!}{k!(n-k)!}$ where $t>0$ integer Thank you
1
vote
2answers
100 views

pascal triangle, getting to the sum of it

So we know that $$\binom{n}0+\binom{n}1+\ldots+\binom{n}n=2^n\;.$$ What about the following sum? $$\binom{n}n+\binom{n+1}n+\ldots+\binom{n+m}n\;?$$ (a) Identify several examples of this sum on ...
1
vote
3answers
30 views

How to show $u_n=\binom{n}{l}\Rightarrow s_n=\binom{n+1}{l+1}$?

Hello respected everyone. Before I ask my query, let us first define binomial coeffcients as follows: For $n, r\in \mathbb N$, we define $$\binom{n}{r}=\begin{cases} \frac{n!}{r!(n-r)!}~~~ ...
1
vote
1answer
37 views

Dealing with floor function in binomial coefficients

I'm trying to estimate $\binom{n}{\left \lfloor{\alpha n}\right \rfloor }$ asymptotically using Stirling's formula. However, I'm a little lost with what to do about the floor function here. In the ...
1
vote
2answers
103 views

Find the number of solutions of the equation $x+y +z +w = 15$ in the following cases:

Find the number of solutions of the equation $x+y +z +w = 15$ in the following cases: (a) $x,y,z,w \geq 0$ (b) $x,y,z,w > 0$ (c) $x>2, y>-2, z>0, w>-3$ I think I have an idea on how ...
1
vote
1answer
30 views

How is the Binomial coefficient simplified to a falling factorial?

I'm learning how to take the derivative of the binomial coefficient and found a blog post that was quite useful. However I am unclear as to how the first step bellow was simplified to the second step ...
0
votes
1answer
38 views

Trailing zeroes in binomial coefficient

I have a doubt regarding trailing zeroes in binomial coefficients... Question: How would you calculate the number of trailing zeroes in the binomial coefficient of ${n\choose r}$ upto values of ...
0
votes
1answer
54 views

Number of ways to combine two sets of values

Good day. I have an algorithm which iterates over a set of values S =[1 -1 0.5 -0.5] for a set of parameters in vector ...
0
votes
1answer
27 views

Asymptotic for binomial coefficients

Does there exist any asymptotic formula for binomial coefficients ${n \choose k}$ for large $n$ when $k$ is fixed?
0
votes
2answers
30 views

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$ where $|z|<1$ and $k \geq 0$. I know The right hand side: \begin{align*} \frac{1}{(1-z)^{k+1}} & = ...
0
votes
1answer
49 views

Remove first element of sequence, compute cumulative sum, iterate

(This is related to my previous question General formula for iterated cumulative sum.) Consider the sequence $S_0$ consisting of ones: $$ 1,1,1,1,1,1,\ldots $$ Now remove the first element and ...
0
votes
1answer
26 views

Integer Series Expansion

For any two $p,q \in \mathbb{Z}$ and $n \in \mathbb{Z}^+$, can one prove that $a_n = \frac{p(-p)^n - q(p-2q)^n}{(p-q)}$ is an integer with recursion relation $a_0 = 1,$ $a_n = ...
0
votes
4answers
37 views

How to computer the summation of a binomial coefficient/ show the following is true

$\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$. I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make ...
1
vote
1answer
33 views

Show that $\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$

The Problem Show that $$\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$$ My Work $$\begin{align*}\frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k} &= ...
0
votes
2answers
33 views

How to show C(n,k)= C(n,n-k)

I am doing a question from the textbook "Calculus and Statistics" by Gemigani. If $n = 2m$ and $k= 1,2, \ldots ,m$ Prove that $C(n,k)= C(n,n-k)$ Ok so my approach begins with writing out the ...
1
vote
1answer
21 views

Binomial series after using binomial identity

Following http://en.wikipedia.org/wiki/Binomial_coefficient#Newton.27s_binomial_series , I am trying to prove that $$ \sum_{\kappa=0}^\infty \binom{\eta + \kappa}{\kappa}x^\kappa = (1 - ...