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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2answers
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What is $\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n}$ for $x > 4$.

What is $\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n}$ for $x > 4$. Here is what I got so far (using Cauchy's integral formula) : $$\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n} ~=~ \...
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1answer
44 views

Proving binomial identities [duplicate]

Can someone help me prove these two binomial identities using either walks in Pascal's triangle or a committee-selection model? $(1)$ $\qquad$ $\displaystyle\sum_{k=0}^m {m\choose k}{n\choose r+k}={m+...
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2answers
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Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I assume....
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2answers
71 views

Probability of winning a prize in a raffle (that each person can only win once)

There is a raffle coming up. 4000 tickets have been sold, and there are 10 prizes to win. I have bought 8 tickets. What are the odds I will win a prize? Note: each person can only win once. There ...
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1answer
46 views

Binomial coefficient of power n

How can I find the coefficient of $x^n$ using binomial theorem? $$\frac{1-x}{(1+x)^3}$$
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1answer
19 views

Self-avoiding walks from one diagonal to the other on $mxn$ lattice is ${m+n \choose m,n} $

According to wikipedia "self-avoiding walks from one end of a diagonal to the other, with only moves in the positive direction, there are exactly $$ \binom{n+m}{n,m} $$paths for an $m × n$ ...
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1answer
17 views

How can I find the cubic polynomial, using $4\times4$ linear system with coefficients? Or any help of reference?

part A in standard form $$y = a_0 + a_1t + a_2t^2 + a_3t^3$$ passes points $$(0, 4), (1, 3), (−1, 7), (2, −2)$$ part B and for the same cubic polynomial in shifted basis $$\{1, t − 2,(t − 2)^2,(...
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1answer
56 views

How many $5$ card poker hands contain at least $1$ red and $1$ black card?

How many $5$ card poker hands contain at least $1$ red and $1$ black card? I used inclusion-exclusion to calculate my answer. The number of total poker card hands are:$$52\choose 5$$I have $26$ red ...
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1answer
43 views

Extending binomial identity $ \sum\limits_{k=0}^n\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$ to $0<x<1$

I found in Matlab that $$ \sum_{k=0}^n~\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$$ for $1\leq x< n$ only (I am about 95% sure of this since the sum is numerically unstable and cannot give ...
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1answer
55 views

Prove $\left(\dbinom nk \right)= \left(\dbinom{k+1}{n-1}\right)$ [closed]

I need to prove $\left(\!\dbinom nk \!\right)= \left(\!\dbinom{k+1}{n-1}\!\right)$ where the double parens denote multiset coefficients and $n,k$ are integers with $1 ≤ k≤ n$ using an algebraic proof. ...
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1answer
29 views

Proof binomial coefficient [closed]

I'm trying to prove the following: $$\binom{n + p}{k} = \sum_{j=0}^n \binom{n}{j} \cdot \binom{p}{k - j}$$ How do I do it? Induction? And can someone hint me at how to start?
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1answer
65 views

Easy geometric sum with binomial coefficient

In the context of stochastic processes I came across the following equality, where $|s| < 1, p \in [0,1]$: $$\sum^\infty_{k=0}(s^2p(1-p))^k\begin{pmatrix} 2k \\ k \end{pmatrix} = \frac{1}{\sqrt{1-...
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3answers
61 views

Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$

Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$. I know he Binomial Theorem will be useful here, but I am struggling to use it with any certainty.
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1answer
22 views

Is there a general formula for the $n$'th variable of the solution for a lower triangular linear system of equations?

I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely ...
2
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2answers
48 views

Using the Binomial Identity, prove that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Using the Binomial Identity, prove that: $${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$$Because this is in the form of a Binomial Coefficient, I can break down the LHS further:$$\left(...
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4answers
32 views

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$ I think I'm having a bit of algebra problem with this proof. Here is my work thus far:$$\frac{n!}{k!(n-k)!}+...
4
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1answer
28 views

Show that $2k\choose k$ divides the lcm of $1, \dots, 2k+1$

I want to show that $(2k+1){2k\choose k}$ is a factor of $\text{lcm}(1, \dots, 2k+1)$. Clearly the divisor is equal to $2^k\frac{1\cdot3\cdot\dots\cdot (2k+1)}{k!}$, but I don't know how to show that ...
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1answer
19 views

calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$

Is there any formula for calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$?
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2answers
211 views

Find the probability of getting one diamond and one spade in a five-card hand, using binomial coefficients.

A five card hand is dealt at random from a standard $52$ card deck. Let $X = \text{# spades}$ and $Y = \text{# diamonds}$. Find $P(X = 1\text{ and }Y =1)$. Leave your answer as a ratio of products of ...
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1answer
28 views

Is this true that $\sum_{v=0}^k (-1)^v{a+v-1 \choose v}{b+k-v-1 \choose k-v}$ is the coefficient of $t^k$ in $(\frac{1}{1+t})^a(\frac{1}{1-t})^b$

I was reading a paper, in which the author assumed that $$\sum_{v=0}^k (-1)^v{a+v-1 \choose v}{b+k-v-1 \choose k-v}$$ is the coefficient of $$t^k $$ in $$\left(\frac{1}{1+t}\right)^a\left(\frac{1}{1-...
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1answer
47 views

Vandermonde's Convolution special case.

I am not able to show this case of Vandermonde's Convolution without using induction. Can someone help me? $$ \binom{n}{m} = \sum_{k=0}^{m} \binom{n-p}{m-k} \binom{p}{k}. $$ I thank now.
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4answers
325 views

Sum of sum of binomial coefficients $\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
4
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1answer
72 views

The sequence $(-1)^n\binom{\alpha-1}{n}$ converges.

I need to show that for $n \in \mathbb N_0$ and $\alpha \ge 0$ the sequence $(-1)^n\binom{\alpha-1}{n}$ converges. It can be shown that the sequence convereges to zero using a theorem claiming that $|...
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38 views

Multivariable recurrence: Solving $c(n,k) = c(n-1,k) + c(n-1,k-1) = \binom{n}{k}$ by algebraic methods.

Let $(c_{n,k})_{n,k=0}^{\infty}$ be defined by $c(0,0)=1$, $\:\:c(0,k)=0 \:\: \forall \: k > 0$ $$c(n,k) = c(n-1,k) + c(n-1,k-1) \:\:\: \forall \: n \geq 1$$ I can show that the answer ...
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1answer
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Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
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2answers
35 views

binomial coefficient where k > n

For solving binomial coefficients we have use from formula $\frac{n!}{k!(n-k)!}$ This formula only works if n > k. What happens if n < k? Is there another formula we need to use?
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6answers
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You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life.

This is from the lecture notes in this course of discrete mathematics I am following. The professor is writing about how fast binomial coefficients grow. So, suppose you had 2 minutes to save ...
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0answers
29 views

arranging $n$ objects of one kind and $m$ objects of other kind in a row

Why are there precisely $\binom{m+n}{n}$ ways of arranging $M$ objects of one kind and $N$ objects of other kind in a row?
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1answer
31 views

Binomial expansion extension to negative powers

I know that: $$\sum_{k=0}^n {n \choose k}a^{n-k}b^k = (a+b)^n$$ But how is this extended to negative powers, for example, I came across the following line of maths, which I struggle to understand: $...
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Logic - Binomial Theorem

I could use some assistance with understanding this problem. I understand that there are ${n}\choose{k}$ is a representation of ${n}\choose{k}$ ways to choose k elements from a set of n elements, ...
3
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0answers
39 views

Find the sum of $\binom{2007}{0}+\binom{2007}{4}+…+\binom{2007}{2004}$ [duplicate]

Find the sum of $$S=\binom{2007}{0}+\binom{2007}{4}+\binom{2007}{8}+...+\binom{2007}{2004}$$ My work so far: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ $$(1-1)^n=0=\...
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2answers
44 views

Alternating series of compositions of triangular numbers

I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls. Each time I add a ball to $S$, it may dislodge another ball with probability proportional to ...
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1answer
27 views

Closed form for binomial sum with absolute value

Do you know whether the following expression has a (nice) closed form or a close enough approximation? $$\frac{1}{2^n}\sum_{k=0}^{n} \binom{n}{k}|n-2k|$$ Thanks a lot :) Cheers, M.
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On a certain series of cosines

For natural numbers $p$ and $q$, compute the value of $$\displaystyle \sum_{k=0}^{q-1} \cos^{p} \left(\dfrac{2\pi k}{q}\right).$$ I got the answer $$\dfrac{q}{2^p} \sum_{l=1}^{p} \binom{p}{l} \...
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2answers
44 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that $f(x)...
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Intuitive explanation of Extended binomial coefficient

We all are familiar with the following formula - $$\dbinom{n}r = \dfrac{n!}{(n-r)! \space r!} \space\space \space ; \space \space n>r$$ This is the binomial formula where $n$ and $r$ are ...
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3answers
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Stirling on ${2n-1 \choose n}$

I'm trying to find an expression for $${2n-1 \choose n}$$ using Stirling's approximation $$k!\sim \sqrt{2\pi k}(\frac{k}{e})^k.$$ I see $${2n-1 \choose n}\approx \frac{1}{\sqrt{2\pi}}\sqrt{\frac{2n-1}...
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1answer
25 views

Generating function for multiset formula

It's said that the generating function for $g(x) = \sum_{d=0}^\infty {d+m-1 \choose m-1} x^d$ is equal to $\frac{1}{(1-x)^m}$. In the proof that I have seen it states that: By the geometric series, $...
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0answers
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Smallest Parameter to Satisfy Exponentially Scaled Binomial Coefficient Inequality

Let $t$ be given, I am mainly interested in large $t$. Define $m(t)$ as below $$ m(t)=\min\left\{m: \sum_{k=0}^m \binom{t+k-2}{k} 2^{t+k} \geq 2^{2t}\right\}. $$ Is there a nice estimate for $m(t)$? ...
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2answers
59 views

Catalan numbers formula derivation

I'm trying to follow a proof of the Catalan numbers being equal to $\frac{1}{n+1} {2n \choose n}$ from the recurrence relation $C_n = C_0C_{n-1}+C_1C_{n-2}+...+C_{n-2}C_{1}+C_{n-1}C_0$ Now it's seen ...
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3answers
39 views

How to sum binomial coefficients which are multiples of 3? [duplicate]

Basically $\sum_{i=0}^{33}\binom{99}{3i} $ I have read about this thread. I am looking for a conventional approach. Like the one we use in $\sum_{i=0}^{n}\binom{2n}{2i} = 2^{2n-1}$
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The number $\binom{8}{4}$ is equal to the number of subsets of size 4 of the set $\{1, \dots, 8\}$

I was asked to proof if is true and give a counter example if it is false. However I prefer True. since all the numbers 1-8 insides the brackets are in the sets. I'm I correct?
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Find the middle number in the $29$th row in the Pascal's Triangle

This question is taken from the Singapore Mathematical Olmpiad training notes for Primary school. Find the middle number in the $29$th row of the Pascal's triangle. For example, the middle number ...
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41 views

Simpler form of binomial coefficients product

I am trying to find a simpler relation or an approximation of a product of binomial coefficients. This product is given by: $\Pi_{i=a}^{N-1}\binom{N}{i+1}$ Or if there is a starting point towards ...
3
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1answer
30 views

Can't Remember a Book about Binomial Sums and Hypergeometry

Some time ago I had come across a website which had the online version of a book about techniques dealing with the solution of sums involving binomial coefficients, and something with the word '...
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0answers
44 views

Calculating summations concerning binominal coefficients

Since $$\sum_{j=0}^{m}\binom{k}{j}\binom{k}{m-j}=\binom{2k}{m},$$ what is the result of $$\sum_{j=0}^{m}\binom{k}{j}\binom{k}{m-j}(\frac{1}{3})^{j}=?$$ Here $i, j, k, m$ are integers.
2
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1answer
44 views

Number of Elements in a Conjugacy Class of $S_N$ (Derivation)

Consider the conjugacy classes of the symmetric group $S_N$. Each conjugacy class consists of permutations that have the same cycle structure. We see that the number of possible cycle structures is ...
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1answer
34 views

Choosing spread out elements

Is there an explicit formula that I'm missing, for the total number of choices from a set $S = \lbrace 1, 2 , \dots , n \rbrace$ such that for every choice $C \subset S$ the following holds: $\forall ...
4
votes
2answers
130 views

Prove that this system of linear equations generates $\left| \left( \begin{matrix} 1/2 \\ n \end{matrix} \right) \right|$ as a solution?

This infinite system of linear equations: $$ \begin{array}( 2x_1=1 \\ 3x_1+4x_2=2 \\ 4x_1+5x_2+6x_3=3 \\ \cdots \end{array} $$ In other words, this is particular case of a system: $$ \begin{array}( ...
2
votes
1answer
45 views

How to prove that $\binom{X+L-1}{L-1} \ge (X-L\times N)^{L-1}$?

I would like to prove the following expression: $$\binom{X+L-1}{L-1} \ge (X-L\times N)^{L-1}$$ , where $X$, $L$ and $N$ are positive integers. Please help me to prove with the following case. $X\ge L\...