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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Compute $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$

I want to calculate $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$. No idea in my mind. Any help? Context I want to calculate the expected value of bits per symbols in adaptive arithmetic coding when the ...
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1answer
23 views

Calculate limit involving binomial coefficient

How can I calculate this limit. Let $p\in[0,1]$ and $k\in\mathbb{N}$. $$\lim\limits_{n\to\infty}\binom{n}{k}p^k(1-p)^{n-k}.$$ Any idea how to do it ?
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15 views

Asymptotic approximate for Binomial sum

How would you approximate the following sum in terms of n: $$\sum_{k=1}^{n} \binom{n}{k} (k-1)(a - n + k)^{-a + n - k - 1/2}(-1 - a - k + 2)^{1 + a + k - 5/2} e^{-n + 2-\frac{(n - k)}{12*a*(a - n + ...
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Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
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1answer
80 views

Prove a theorem in combinatorics

I want to show that for $k=1,...,(n-1)$ we have : $\binom{n}{k}\leq \frac{n^n}{k^k(n-k)^{n-k}}$ I have used induction on $k$, but I have not deduced the above relation.
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+50

An asymptotic expression of sum of powers of binomial coefficients.

Let $k$ be a fixed positive number and $n$ an integer increasing to infinity. Then $$\sum_{\nu =0}^n \binom{n}{\nu}^k \sim \frac{2^{kn}}{\sqrt{k}} \left( \frac{2}{\pi n} \right)^{\frac{k-1}{2}}.$$ ...
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0answers
22 views

Binomial transform

how can be prove expression: $\sum \limits_{s = 0}^{2k} (-1)^s\binom{n+s}{n}\binom{n+2k-s}{n} = \binom{n+k}{k}$ by using this identity: $(1 − t)^{−n−1}(1 + t)^{−n−1}= (1 − t^2)^{−n−1},$ or how ...
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1answer
34 views

Showing the equality of two rook polynomials.

I'm reading Barbeau's Polynomials. I've done the following: Taking an arbitrary chessboard $C$ with some of the squares forbidden (with $n$ being the number of squares and $F$ the number of ...
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1answer
75 views

Find asymptotics in a given form $n=(e+o(1))^{f(s)}$

Let $p\to\infty$, $s={\binom {p^4} p}$ and $n={\binom {p^4}{p^2}}$. Find a function $f(s)$ in the following form $$\large n=(e+o(1))^{f(s)}$$ I've tried to use the followinf asymptotics for ...
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4answers
26 views

Help in proving an algebraic identity involving powers of binomials.

For some reason I found this equation: $(1 + x)^n - 1 = x \sum\limits_{k=0}^{n-1} (1+x)^k$ I think that this is an identity. If for instance one expands the powers and the sum for n = 4, the ...
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8answers
350 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
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2answers
59 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
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7answers
161 views

$\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$

Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to ...
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0answers
38 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
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1answer
25 views

Proving the binomial coefficients by induction (half-done, but need help)

Defining the binomial coefficients $n \choose k$ as follows, i) for all $n \in \mathbb{N}$, $\binom{n}{0} = \binom{n}{ n} = 1$ (ii) for all $2 \leq n \in \mathbb{N}$ and for all $ 1 \leq k \leq n-1, ...
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1answer
56 views

Proving a certain sum is 1 [duplicate]

I'd like to prove that, for any $M,N\in\mathbb{N}$ with $M\leq N$, and any $n\in\mathbb{N}$ with $n\leq M$, the sum: $$\sum\limits_{k=0}^n\frac{\binom{M}{k}\!\!\binom{N-M}{n-k}}{\binom{N}{n}}=1.$$ I ...
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1answer
32 views

Maximum $a \in (0,1)$ such that sum of binomials in $[(n/2)-n^a,(n/2)+n^a]$ is $o(2^n)$

Is the maximum value $a$ such that the sum of the middle binomial coefficients $$2^{-n} \sum_{k=(n/2)-\lfloor n^a \rfloor}^{(n/2)+\lfloor n^a \rfloor} C_k^n$$ goes to zero known?
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3answers
73 views

Find $k$ given that ${14 \choose k} = {14 \choose k-4}$

Ok, so I stumbled upon the question on the title these days, when going over Apostol's Calculus I. Now, because of the placement of the question in the exercises section, I'm convinced that the book ...
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1answer
14 views

Binomial Coefficient as Sum of a Sum

Few days ago, I found this equation: $ \sum_{i=1}^n \sum_{j>i} \frac{1}{2} = {n \choose 2} \frac{1}{2} $ I didn't manage to prove it. Does anyone of you know how to prove it?
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281 views

Prove that $\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$

Prove that: $$\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$$ i know that: $$\sum_{k=0}^n {n \choose k} = {2^n}$$ how to get the (n + n^2)?
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2answers
50 views

The smallest $n$ for which the sum of binomial coefficients exceeds $31$

I have a problem with the binomial theorem. What is the result of solving this inequality: $$ \binom{n}{1} + \binom{n}{2} + \binom{n}{3} + \cdots +\binom{n}{n} > 31 $$
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4answers
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Proof that $\sum\limits_{k = 0}^n {n\choose k} =2^n$ using Binomial Expansion Formula

HW problem here. Not sure how to even start on it. Prove that $$\sum\limits_{k = 0}^n {n\choose k} =2^n$$ Any help is appreciated. For Search purposes: (Hint: Use the binomial expansion mentioned ...
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2answers
123 views

How many Binary numbers?

How many binary numbers of length $n$ can be generated where $n > 7$ and the number either start with $000$ or end with $111$? My questions is, can I choose an $n$ randomly? For example, let's say ...
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1answer
98 views

Closed form of $\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$

Is there a closed form for: $$\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$$ where: $$h(x)=(1-x)^{\alpha}(A-Bx)^{\frac{1}{\gamma}-\alpha}$$ and ...
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2answers
69 views

combinatorial or algebraic proof of combinatorial identity

I would like to find out how to prove the following identity, assuming it is correct: $\displaystyle\sum_{r=0}^n\binom{n}{r}\binom{m+r}{l}=\sum_{r=0}^n\binom{n}{r}\binom{m}{r+l-n}2^r$ for ...
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1answer
22 views

Probability problem (Bernoulli trial)

I recently became interested in studying probability and I stumbled upon this question: There are three points: A, B and C. Exactly two paths exist between A and B and exactly two paths exist between ...
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3answers
38 views

Explanation about an identity involving inverse binomial coefficients.

Now, I was solving a this problem. It asks for summation of $$\sum\limits_{k =0}^\infty\dfrac{1}{{n+k \choose n}}$$ I solved it using this answer, the answer turns out to be $$\dfrac{n}{n-1}$$ ...
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1answer
71 views

Find summation of following series.

What will be the formula for following infinite series? $$1 + \frac{1!}{x+1} + \frac{2!}{(x+1)(x+2)}+ \cdots$$ $$ x\ge2 $$ up to infinite What pattern i got : coefficient of $ \frac{1!}{x+1}$ ...
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1answer
47 views

Proving a Binomial Identity

Can you please help me with problem 25. I need to prove that $f(n+1)=2 f(n)$, where $f(n)$ is the LHS of the expression, from there on I can do it my self. I have tried using the binominal theorem ...
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28 views

Find $f(n)$ in $\binom {2^n} {n^4} = (f(n)+ o(1))^n$

Task is to find $f(n)$ in the following equation: $\binom {2^n} {n^4} = (f(n)+ o(1))^n$ I've found that the problem is a bit over my head. I'm attaching my partial solution below: With use of the ...
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54 views

find the the variable that maximizes a function

I have a function that I am trying to find for what input it maximizes. $$ f(n) = {\binom{S}{2}}^{n/S}$$ I need to find the $S$ for which this function maximizes (for infinite $n$). more generally, ...
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1answer
35 views

Growth of ratio of binomials polynomial or exponential?

Is the growth of $$ \dfrac{\binom{2n}{\sqrt{n}}}{\binom{n}{\sqrt{n}}} $$ polynomial or exponential (or other kind of growth) in $n$? I tried using the Stirling's approximation, which gives ...
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4answers
4k views

How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
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1answer
34 views

Evaluation of a limit of ratio of sums [closed]

How do I calculate the value of $$ \lim_{n\to \infty} \left(\frac{\sum_{r=0}^{n} \binom{2n}{2r}3^r}{\sum_{r=0}^{n-1} \binom{2n}{2r+1}3^r}\right)$$
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4answers
213 views

Alternating sum of binomial coefficients multiplied by (1/k+1)

I'm trying to prove that $$\sum_{k=0}^n {n \choose k} (-1)^k \frac{1}{k+1} = \frac{1}{n+1}$$ So far I've tried induction (which doesn't really work at all), using well known facts such as ...
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2answers
24 views

Proofs of identity for product of binomial coefficients

While verifying my answer to another question, I came across a problem of binomial coefficients: Does $\hspace{.2cm}\displaystyle \prod_{k=1}^{n-1}\binom{n-1}{k}=\prod_{k=1}^{n-1}k^{2k-n}$ for all ...
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1answer
73 views

Proof of an identity involving binomial coefficients

I have found numerically that the following identity holds: \begin{equation} \sum_{n=0}^{\frac{t-x}{2}} n 2^{t-2n-x}\frac{\binom{t}{n+x}\binom{t-n-x}{t-2n-x}}{\binom{2t}{t+x}} = ...
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2answers
76 views

Finding the coefficient of a generating function

Given $f(x) = x^4\left(\frac{1-x^6}{1-x}\right)^4 = (x+x^2+x^3+x^4+x^5+x^6)^4$. This is the generating function $f(x)$ of $a_n$, which is the number of ways to get $n$ as the sum of the upper faces of ...
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How to calculate $\sum\limits_{k=0}^{n}{k\dbinom{n}{k}}$ [duplicate]

I derived this sum from a problem I have been working on. Somehow I don't know how to proceed. I only know some basics like $\sum\limits_{k=0}^{n}\dbinom{n}{k} = 2^n$. Meanwhile I am reading the ...
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1answer
411 views

Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?

A known identity of binomial coefficients is that $$ \sum_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}. $$ Is there a combinatorial proof/explanation of why it holds? Thanks.
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1answer
29 views

Product of binomial coefficients

Is there any way to simplify given expression ($j$ and $i$ are given, $n\leq \lfloor j/i \rfloor$) $$\prod_{x=1}^n \binom {j-(x-1)i} {i}$$ (e.g. in terms of factorials)? Thanks!
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1answer
56 views

prove that $ \binom n 2 + \binom {n-2} 2 + \binom {n-4} 2 + \dots + \binom 3 2 = \frac 1 {24} (n-1)(n+1) (2n+3) $

$$ \binom n 2 + \binom {n-2} 2 + \binom {n-4} 2 + \dots + \binom 3 2 = \frac 1 {24} (n-1)(n+1) (2n+3) $$ where n is odd. Plesase help mi with that equation.
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32 views

Black bears and tan-colored bears catching salmon in Alaska

One of popular tourist attractions in Alaska is watching black bears catch salmon swimming upstream to spawn. Not all "black" bears are black, though- some are tan-colored. Suppose that 6 black bears ...
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0answers
51 views

Permutation and combination identity

Prove that $\displaystyle \sum_{i=0}^n \binom{n}{i}\binom{m+i}{n}=\sum_{i=0}^n \binom{n}{i}\binom{m}{i} 2^i$ for natural numbers $m,n.$ The question doesn't seem to have any direct combinatorial ...
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2answers
23 views

How can I solve for $n$ in this binomial coefficient equation

How can I solve for $n$ in this binomial coefficient equation? $${n\choose 3} = {n\choose 9}$$ When I try to expand it using factorials, I get a very, very long equation, involving $n-s$ up to $n^6$ ...
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3answers
48 views

combinational proof for $ 1 + 2 + \cdots + n = \binom {n+1} 2 $

$$ 1 + 2 + \cdots + n = \binom {n+1} 2 $$ Please give me a help with combinational proof for this formula. Greetings for everybody and thanks in advance.
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1answer
70 views

World series lengths competition, binomial distribution.

Listed in the following table is the length distribution of World Series competion for the 58 series from 1950 to 2008 (there was no series in 1994). WORLD SERIES LENGTHS (note, the total = 58) of ...
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2answers
27 views

counting the rectangles in nxn square

How many different rectangles can be seen in an $$ n \times n $$ grid like the one shown? Of course the rectangles must be at least one box wide and deep, and squares are allowed. ...
2
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2answers
2k views

How do I determine the possible number of combinations of two ordered sets?

I'm not quite sure what the mathematical term for what I'm asking is, so let me just describe what I'm trying to figure out. Let's say that I have two ordered sets of numbers $\{1, 2\}$ and $\{3, ...
2
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1answer
77 views

Proof with combinatorial argument

Show with combinatorial argument that this is equal : $$\dbinom{n}{k+1} = \dbinom{n-1}{k}+ \dbinom{n-2}{k} +...+ \dbinom{k}{k}$$ I have no idea how to do that so it would be really helpful ...