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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How to evaluate: $\int_0^1x^{n-1}(1-x)^{n+1}dx$

How can I evaluate the following integral? ($n \in R$, $n>0$) $$\int_0^1x^{n-1}(1-x)^{n+1}dx$$ I was solving the following problem (as practice) in school: Prove that the sum of $n+1$ terms of ...
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1answer
35 views

The sum of binomial coefficients $\sum_{i=1}^n \binom{i}{2}= {n+1 \choose 3}$

Prove by induction: $$\sum_{i=1}^n \binom{i}{2}= {n+1 \choose 3}$$ I already know that: $$\sum_{i=1}^n \binom{i}{2} = {i+1 \choose 2+1}$$ And the LHS is now equal: $$\sum_{i=1}^n \binom{i}{2} + ...
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prove that $(\frac{1}{6})^{4}\cdot\lim_{n\rightarrow\infty}\sum_{i=4}^{n}\binom{i-1}{3}(\frac{5}{6})^{i-4}=1$

I have to prove the following: $(\frac{1}{6})^{4}\cdot\lim_{n\rightarrow\infty}\sum_{i=4}^{n}\binom{i-1}{3}(\frac{5}{6})^{i-4}=1$ any ideas? thanks
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1answer
34 views

GCD of binomial coefficients of the form ($p^n$ choose $k$)

Let $n$ be a positive integer and $p$ be a prime. Find the greatest common factor of $\binom{p^n}{1}, \binom{p^n}{2},...,\binom{p^n}{p^n-1}$. Progress: We know that for any given $n$ and $k$ in ...
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1answer
38 views

Equating coefficients of binomial expansion modulo p

In this answer: http://math.stackexchange.com/a/652909 Ted equates mod $p$ the coefficients of $$\sum_{n=0}^{pa} \binom{pa}{n} x^n$$ and $$\sum_{i=0}^{a} \binom{a}{i} x^{pi}$$ to get that ...
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1answer
139 views

Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$

How to find this alternating sum of binomial coefficients? $$\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$$
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1answer
44 views

In how many ways can four persons each throwing dice once sum up to 13?

I am solving it by finding out Coefficient of $x^{13}$ in $(x+x^2+....x^6)^4$ but I cannot get the correct answer. Please provide me the final answer if method I am following is correct.
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2answers
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relationship between pascal's triangle and number of combinations?

I was able to solve a classic algorithm question, robot paths by using pascal's triangle (PT). This is where a robot starts in the upper left corner and can only go down or right. I kind of reverse ...
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Determine maximal addend in Newton Binomial Expansion.

Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$ ...
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1answer
37 views

Are there four consecutive binomial coefficients in a row in an arithmetic progression?

Are there four consecutive binomial coefficients in a row in an arithmetic progression? This is suggested by Will Jagy's comment to this question: Find $n$ and $k$ if ...
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Find $n$ and $k$ if $\:\binom{n\:}{k-1}=2002\:\:\:\binom{n\:}{k}=3003\:\:$

$$ \:\binom{n\:}{k-1}=2002\:\:\:\binom{n\:}{k}=3003\:\: $$ What are the values for n and k? My initial idea was to divide those two: ...
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2answers
34 views

how to find the coefficient of $x^t$ in multiplication of series

how to find the coefficient of $x^5$ in $$(1+x+x^2+x^3+.....)(1+x^2+x^4+x^6+.....)(1+x^3+x^6+x^9+....)(1+x^4+x^8+....)$$ Is there any method in general to find the coefficient of $x^t$ in the above ...
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1answer
43 views

Convergence of a series with binomial coefficient

Let $S_n$ be a series with binomial coefficients as follows: $$ S_n = \sum_{k=1}^n \begin{pmatrix} n \\ k \end{pmatrix}\frac 1k\left(-\frac{1}{1-a} \right)^k\left(\frac{a}{1-a} \right)^{n-k},$$ where ...
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214 views

Complicated sum with binomial coefficients

I know how to prove, that $\frac{1}{2^{n}}\cdot\sum\limits_{k=0}^nC_n^k \cdot \sqrt{1+2^{2n}v^{2k}(1-v)^{2(n-k)}}$ tends to 2 if n tends to infinity for $v\in (0,\, 1),\ v\neq 1/2$. This can be proved ...
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What does ${50}\choose{4}$ mean in statistics?

I have a test tomorrow in statistics and was wondering what the following means? $$\binom{50}{4}$$ My professor along with most of my classmates have a calculator they can just plug that into. The ...
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3answers
107 views

How to prove that $\frac1{n\cdot 2^n}\sum\limits_{k=0}^{n}k^m\binom{n}{k}\to\frac{1}{2^m}$ when $n\to\infty$

I have solve following sum $$\sum_{k=0}^{n}k\binom{n}{k}=n2^{n-1}\Longrightarrow \dfrac{\displaystyle\sum_{k=0}^{n}k\binom{n}{k}}{n\cdot 2^n}\to\dfrac{1}{2},n\to\infty$$ ...
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Fractals with Moduli in Pascal's Triangle

I'm working through a problem for my graduate math class and am hitting a wall. Here's the problem: For the first 10 lines of Pascal's Triangle, replace the odd numbers by black squares and the even ...
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161 views

Is the numerator of $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\binom{n}{k}$ a power of $2$?

I stumbled on something numerically, and was just starting to work on it, but it seemed fun enough to share. Let $$f(n)=\sum_{k=0}^{n} \frac{(-1)^{k}}{2k+1}\binom{n}{k}$$ It appears, from the ...
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1answer
132 views

A summation series of binomial coefficients

Given. $$(1 + x^{2005} + x^{2006} + x^{2007})^{2008} = A_0 +A_1x +A_2x^2 + \cdots + A_nx^n$$ We are required to calculate $A_0 - A_1/3-A_2/3 + A_3 -A_4/4 -A_5/5 + A_6 - \cdots$ I tried approaching ...
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71 views

Find missing numbers given the sum

Sam was given a task to add up numbers 1 + 2 + 3... up to ten. As he stopped, gave a sum of 37. It was declared that his results are wrong. Later he discovered that he had missed some numbers in the ...
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47 views

Represent $f(x)=\frac{1}{(1-x^2)^4}$ as a power series

Represent $f(x)=\frac{1}{(1-x^2)^4}$ as a power series $$\sum\limits_{n=0}^{\infty}x^{2n}=\frac{1}{1-x^2}$$ Second derivative is $$\left(\frac{1}{1-x^2}\right)^{''}=\frac{1}{(1-x^2)^4}\cdot ...
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2answers
57 views

Prove equality between binomial coefficients.

Using the Binomial theorem, prove that: $$ \binom{m+n}{k}=\sum_{j=0}^k \binom{n}{j}\binom{m}{k-j},\; 0\leq k\leq m+n$$
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When $\frac{1}{n}\binom{n}{r}$ is an integer , again?

This question follows a previous one If $n$ and $r$ are coprime then $a_{n,r}=\frac{1}{n}\binom{n}{r}$ is integer but this is not a necessary condition. Question: what is a necessary and ...
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Inequalities involving binomial coefficient when n varies

I am trying to invoque some inequality of the form: $$ {n+i \choose k} \leq f\left ({n \choose k},n,k \right) $$ where $i$ is small (typically 1, 2 or 3). The tighter the better, however any ...
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1answer
21 views

Random variable and probability distribution: (5 men and 5 women are ranked)

An exercise in my homework: 5 women and 5 men are ranked based on their results in a test. Assume that all results are different and every of 10! possible orders has the same probability. X (random ...
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36 views

Prove the equality

Prove the equality $\sum\limits_{k=0}^{n-1}{4n\choose 4k+1}=2^{4n-2}$ $$\sum\limits_{k=0}^{n-1}{4n\choose 4k+1}=2^{4n-2}=\frac{1}{2}\sum\limits_{k=0}^{n-1}{4n\choose ...
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113 views

when is $\frac{1}{n}\binom{n}{r}$ an integer

So I am considering for which values of n is $a_n =\frac{1}{n}\binom{n}{r}$ an integer for all $ 1\leq r \leq n-1 $. The first thing I did was to check the Pascal Triangle. So I guess n has to be ...
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1answer
30 views

Expand generating function $\frac{\exp{\frac{z}{1-z}}}{1-z}$

I know that the coefficients $[z^n]$ of the exponential generating function $\frac{\exp{\frac{z}{1-z}}}{1-z}$ are $\sum_{i=0}^{n}i!\binom{n}{i}^2$ but have trouble in proving it. I have done the ...
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1answer
117 views

Prove combinatorial Identity using a combinatorial argument. [duplicate]

I have proved by induction on k that : $$ \binom{n}{n} + \binom{n+1}{n} + \binom{n+2}{n}+ ... + \binom{n+k}{n} = \binom{n+k+1}{n+1} $$ But now, I need to prove it using a combinatorial argument. ...
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1answer
54 views

Closed form of to calculate variation of Pascal's triangle

So if you have Pascal's triangle, I know you can calculate any value in closed form. 1 1 1 1 2 1 1 3 3 1 .... If we let ...
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2answers
93 views

Prove the identity $\binom{-x}{k}=(-1)^k\binom{x+k-1}{k}$ for complex number $x$

Prove that for all complex numbers $x$ and all $k \in \mathbb{N}$ we have $$\binom{-x}{k}=(-1)^k\binom{x+k-1}{k}$$ The fact that we have a complex number in the identity confuses me, because I ...
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Prove the identity $\sum^{n}_{k=0}\binom{m+k}{k} = \binom{n+m+1}{n}$

Let $n,m \in \mathbb{N}$. Prove the identity $$\sum^{n}_{k=0}\binom{m+k}{k} = \binom{n+m+1}{n}$$ This seems very similar to Vandermonde identity, which states that for nonnegative integers we ...
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33 views

Determine the real coefficients of a polynomial

I need to determine the real coefficients $a, b, c$ of the following polynomial: $$P(x) = x^5 + ax^4 - 2x^3 - 6x^2 + bx + c$$ I know that $P(-2) = 9$, and the sum of the solutions (roots of the ...
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2answers
91 views

Evaluate sum using generating function

I'm trying to evaluate this sum: $$ \sum\limits_{s = 0}^{500} (-1)^s \binom{3000 - 2s}{2000} \binom{2001}{s}$$ As I think we need to use an expansion of $(1 -x)^n (1+x)^k$, but I've tried several ...
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2answers
99 views

Proving identities using combinatorial interpretation of binomial coefficients

Let $n \in \mathbb{N}$. Prove the identities $$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$$ and $$\sum^{n}_{k=0}\binom{n}{k}^2 = \binom{2n}{n}$$ by using only the combinatorial interpretation ...
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1answer
120 views

Mathematical induction $\sum_{i=0}^{n}{n \choose i} = 2^n $

Prove by mathematical induction: $\sum_{i=0}^{n}{n \choose i} = 2^n ; n \ge 0$ Step 1: n = 0 ${0 \choose 0}=2^0$ Step 2: for n = k $\sum_{i=0}^{k}{k \choose i} = 2^k$ assumption: for n = k+1 ...
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Understanding steps in a proof of $\sum_{k=1}^n\binom{n-1}{k-1}\binom{2n-1}{k}^{-1}=\frac{2}{n+1}$

So, the task is to prove that: $$\sum_{k=1}^n\binom{n-1}{k-1}\binom{2n-1}{k}^{-1}=\frac{2}{n+1}$$ I tried different methods but none led me to the solution. I looked up the solution and I can't even ...
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1answer
27 views

Even numbers in Pascal's triangle.

Basically i've been looking at Pascal's triangle and been wondering how it represents Sierpinski's triangle once the even numbers are shaded. Once I rewrote the triangle in terms of C I observed that ...
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1answer
187 views

Binomial coefficients that sum to a perfect square

I was wondering about a property of a sum I recently saw. $${8\choose2}+{9\choose 2}+{15\choose2} + {16\choose2}=17^2$$ And if we increment the terms $${9\choose3}+{10\choose 3}+{16\choose3} + ...
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1answer
35 views

Can you find the coefficient of x^r in a generating function if r is a negative number?

Find the coefficient of $x^8$ in $(x^2 + x^3 + x^4 + x^5)^5$. I pulled the $x^2$ out to make it $$\left[x^2(1 + x + x^2 + x^3)\right]^5$$ and then $$x^{10}(1 + x + x^2 + x^3)^5$$ But then ...
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Sum of the rows of Pascal's Triangle.

I've discovered that the sum of each row in Pascal's triangle is $2^n$, where $n$ number of rows. I'm interested why this is so. Rewriting the triangle in terms of C would give us $0C0$ ...
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1answer
32 views

Proportion with 3 variables - binomial coefficients

So, if we're given something like this: $$\binom{n}{k}:\binom{n+1}{k}:\binom{n+1}{k+1}=3:4:8$$ How do I rewrite this so I can manipulate it? Edit: Is there a general procedure for n variables?
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Name of statistickal method?

In an assignment, I tested if $H_0$ and $H_A$ were true. I had to use the variables as binominally distributed, since I only had ordinal data to work with. Apparantly, this method has a special name, ...
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1answer
28 views

Problem with binomial coefficients and their symmetry

I'm going through my workbook and ran across this: $$\sum_{k=0}^{n-1}\binom{4n}{4k+1}=\frac{1}{2}\sum_{k=0}^{n-1}\binom{4n}{4k+1}+\frac{1}{2}\sum_{k=0}^{n-1}\binom{4n}{4(n-k-1)-1}$$ ...
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1answer
52 views

Combinatorial proof of $1=\sum_{i=0}^n{(-1)^i \binom{n}{i} 2^{n-i}}$

I'm searching for a combinatorial proof of the following equality: $$1=\sum_{i=0}^n{(-1)^i \binom{n}{i} 2^{n-i}}$$ It's trivial to show using Newton's binomial theorem: $(-1+2)^n=1$, but I'm ...
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1answer
32 views

Proving an inequality which looks like we could use Bernoulli's inequality

How can we prove this inequality: $$\left(1+\frac{1}{n}\right)^n<3$$ What I did is: $$(1+\frac{1}{n})^n=\sum_{k=0}^{n}\binom{n}{k}1^{(n-k)}\frac{1}{n^k}=$$ ...
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3answers
52 views

Show that $\frac{n(n-1)(n-2)\times\cdots\times(n-r+1)}{r!}=\frac{n!}{r!(n-r)!}$ = The binomial coefficient formula

I have written in a textbook that $$\cfrac{n(n-1)(n-2)\times\cdots\times(n-r+1)}{r!}\tag{1}$$ $$=\cfrac{n(n-1)(n-2)\times\cdots \times 2 \times 1}{r!(n-r)(n-r-1)\cdots \times 2 \times 1}\tag{2}$$ ...
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1answer
53 views

Find the Coefficient of $x^3$ in $(2+x^2)^3 (3+2x)^7$

I'm asked to find the coefficient of $x^3$ in $(2+x^2)^3 (3+2x)^7$. For a simple problem like finding $x^2y^3$ in $(x+y)^5$ I can solve easily using binomial theorem. But I have no idea how to go ...
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1k views

An identity for the factorial function

A friend of mine was doodling with numbers arranged somewhat reminiscent of Pascal's Triangle, where the first row was $ 1^{n-1} \ 2^{n-1} \dots n^{n-1} $ and subsequent rows were computed by taking ...
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3answers
30 views

Binomial expansion to find a specific term (coefficient)

for this question I tried to use binomial theorem to find a specific term. However, I eventually cannot find a valid value of n and r and p. My working is shown in the picture and please tell me my ...