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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Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$

I need hint proving this $$\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$$
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2answers
27 views

negative binomial distribution problem

Find the probability that you find 2 defective tires before 4 good ones. There is a chance of a tire being defective at a rate of 5%. From my understanding with the negative binomial distribution we ...
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2answers
39 views

Expansion Coefficient needed

This is probably something very easy, but wth... my mind is totally stuck right now. I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$ Well I know that the answer ...
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2answers
154 views

Alternating sum of binomial coefficients $\sum(-1)^k{n\choose k}\frac{1}{k+1}$ [duplicate]

I would appreciate if somebody could help me with the following problem Q:Calculate the sum: $$ \sum_{k=1}^n (-1)^k {n\choose k}\frac{1}{k+1} $$
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3answers
427 views

Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$?

Which is greater among $300 !$ and $(300^{300})^\frac{1}{2}$ ? The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.
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4answers
111 views

Find $\sum_{r=0}^{2n}\frac{r}{r+2}\binom{2n}r\;.$

Find $$\sum_{r=0}^{2n}\frac{r}{r+2}\binom{2n}r\;.$$ I got $$\frac{2^{2n+1}(2n^2+n+1)-1}{(2n+1)(2n+2)}$$ but the answer is $$\frac{2^{2n+1}(2n^2-n+1)-2}{(2n+1)(2n+2)}$$ Thanks for the help...
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0answers
76 views

Combinatorial Identity

I have to validate the following identity which is defined: $$ \sum_{k=1}^n (-1)^{k-1}*q^{\frac{k(k-1)}{2}} *\frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1 $$ where $0<q<1$. I ...
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1answer
61 views

Combinatorics - finding coefficients when summing over permutations of permutations

I have $N$ 2-tuples. Each tuple* can either be up, in which case it has components $(a,b)$, or it can be down, in which case it has components $(c,d)$. Given that exactly $N_\mathrm{up}$ of these ...
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0answers
29 views

Analytical solution for binomial equation

Suppose that the random variable $X \sim \operatorname{Binomial}_{n,p}$, and suppose we have $p' \in [0,1]$. I have been asked to solve for the least $n$ such that $P(X \leq 2) = p'$. It was ...
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2answers
304 views

Binomial Identity $\sum\binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$

I'm looking for a reference with the proof of the following binomial identity: $$\sum_{k=0}^n \binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$$ I've looked in a number of textbooks that have a ...
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1answer
36 views

combinatorics - number of ways to choose r out of n (with inclusion-exclusion)

Quick question. Out of a set of $n$ apples, we are given that $m$ are delicious. Show that the number of different combinations to choose $r$ apples that contain all of the delicious ones is given by ...
3
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2answers
68 views

combinatorics - fixed point permutations

Simple question but I just need a little tip to finish it. we are given $A=\{1,2,3...,2n-1,2n\}$ the set of all integers between and including $1$ and $2n$. We are asked how many different ...
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3answers
411 views

Prove the lecturer is a liar…

I was given this puzzle: At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are ...
3
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1answer
87 views

Pascal Triangle general formula

I'm working on a presentation on the Binomial Theorem for my Algebra 2 class and while writing Pascal's Triangle, I came across one of the properties that I haven't seen in a while. That being ...
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1answer
45 views

Two team playoff question, homework help

I am working through my college text book (mathmatical statistics freund/walpole) trying to refresh my stat skills. Its been a couple years... I would sure appreciate any pointers on an exercise ...
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1answer
64 views

Relation between Binomial coefficient and Stirling number of second type

Is that true, that for every n,k such that $$k>1$$ we have the inequality $${n \choose k} \leq {n \brace k}$$?
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3answers
120 views

How many numbers $k$ of $200 \choose k$ are divisible by $3$? $k \in \{0,1,2,\cdots 200\}$

"How many of the numbers (200 Choose k), where k is an element of the set {0,1,2,3,4,....,200} are divisible by 3? " Here is my thinking: (200 Choose 0,1, and 2) are not multiples of 3 but every ...
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1answer
50 views

Express number of ways integer can be written as coefficient in generating series

Question: "Express the number of ways that an integer $n$ can be written as a sum of a cube of an integer $s\ge-1$ plus the fourth power of an integer $t$ plus the square of an odd integer $r$ as a ...
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2answers
107 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
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64 views

Combinatorial word problems (Discrete math)

I have a problem with writing the word problems to which the answers are the following expressions. I am not sure if these answers sound right. I am not good with writing questions to these ...
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0answers
34 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
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2answers
169 views

Closed form expression for unusual sum of binomial coefficients

How do I get a closed form expression for $\sum_{i=c}^{n} i\binom{i}{c}$? Note that the index ranges over the upper values of the binomial, not the lower. I know computer algebra systems can give me ...
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1answer
80 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
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2answers
46 views

series sum of binomial co-efficients

$C_r$ stands for $_nC_r$ We have to show that $ \frac{C_0}{1} -\frac{C_1}{5} + \frac{C_2}{9} +\ldots+ (-1)^n\cdot\frac{C_n}{4n+1} = \frac{4^n\cdot n!}{1.5.9\ldots(4n+1)}$ What I have done : ...
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1answer
45 views

Binomial distribution . Heads and Tails

Consider a coin with P(Heads) = 2/ 3 . We toss this coin 100 times (assume that the tosses are independent). Determine the probability that we get exactly 45 tails out of the 100 tosses. First, ...
10
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1answer
181 views

Strehl identity for the sum of cubes of binomial coefficients

In 1993 Strehl showed that $$ \sum_k\binom nk^3=\sum_k\binom nk^2\binom{2k}n. $$ I’m interested in a combinatorial proof. Upd (Jan '14). Maybe the original question was too restrictive — I'm now ...
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0answers
23 views

Shortest ways in a grid above the angle bisector

Suppose, you have a grid with the side lengths n and m and the angle bisector from the upper left corner to the bottom side. To walk along the lines from A to B, there are $\binom{n + m}{n}$ shortest ...
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2answers
75 views

Combinatorial proof with binomial coefficients

I need to prove this with combinatorial arguments. I don't know how to start. $$ \sum_{j = r}^{n + r - k}{j - 1 \choose r - 1}{n - j \choose k - r} = {n \choose k}\,, \qquad\qquad 1\ \leq\ r\ \leq\ ...
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0answers
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Bound for squarefree binomial coefficients

Numerical tests up to $x=2000$ seem to suggest that the number of squarefree binomial coefficients is bounded at $\dfrac{112x^{\ y}}{239^{\ y}}$: where ...
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1answer
97 views

Squarefree binomial coefficients.

At $n=23$, all binomial coefficients are squarefree. Is this the largest value for $n$ where this is the case? Edit A plot up to $n=50$: A plot up to $n=500$: plotted against $n+1$ and ...
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1answer
25 views

List of length $x$, with letters $A$ and $B$, how many ways? (Basic Combinatorics Question)

I was tasked with finding the number of possible ways of writing a sequence with the following conditions: Sequence has a length $x$, where $x$ is even Sequence consists of the letters $A$ and $B$ ...
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8answers
274 views

Show that $ \sum_{n=2}^m {\vphantom{+1}n \choose 2} = {m+1 \choose 3}$

I need a hand in showing that $$ \sum_{n=2}^m {n \choose 2} = {m+1 \choose 3}$$ Thanks in advance for any help.
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4answers
58 views

${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$

I need to prove that $${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$$ I started by writing out all the terms using the formula ${n!\over{k!(n-k)!}}$ but I can't make the two ...
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1answer
50 views

Why is $ 2\binom nm^2<n^{2m}$?

$\forall n\geq2 \forall m\geq2,$ $$ 2\binom nm^2<n^{2m}.$$ Why is the above inequality, which is equivalent to $ \binom nm<\frac{n^m}{\sqrt 2}$, true?
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3answers
69 views

Why is this binomial coefficient bounded thus?

Source: Miklos Bona, A Walk Through Combinatorics. $$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$ The RHS is the upper bound of the Ramsey number $R(k,k)$. How can I prove the inequality ...
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0answers
63 views

Is there a closed form formula for the Bernoulli numbers?

A while ago I found this algorithm. Today I read in wikipedia that Euler zig zag numbers can be used for computing the Bernoulli numbers. This Mathematica program computes the Euler zig zag numbers ...
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2answers
95 views

Evaluate $\sum\limits_{k=2}^n \frac{n!}{(n-k)!(k-2)!} $

Question is to Evaluate $$\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!} $$ What i have done so far is $$\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!}=n(n-1)\sum_{k=2}^n \frac{(n-2)!}{(n-k)!(k-2)!}=n(n-1)\sum_{k=2}^n ...
4
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2answers
85 views

Proofs from the Book - need quick explanation

I've been recently reading this amazing book, namely the chapter on Bertrand's postulate - that for every $n\geq1$ there is a prime $p$ such that $n<p\leq2n$. As an intermediate result, they prove ...
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1answer
54 views

Maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$

Is there an expression for the maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ (i.e. $\max_{k\in\{0,\ldots,n\}}{n\choose k}\lambda^k)$ in terms of elementary functions of $n$ and ...
2
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2answers
60 views

Why is $\binom{a}{n}=(-1)^{a}\frac{\sin(n\pi)}{(a+1)\binom{n}{a+1}\pi}$?

Why is $\displaystyle\binom{a}{n}=(-1)^{a}\frac{\sin(n\pi)}{(a+1)\binom{n}{a+1}\pi}$? (A particular case popped up as an alternative formulation in WolframAlpha while operating with binomials. Any ...
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2answers
44 views

prove that for $n \ge 4, {{2n}\choose{n}} \ge n\cdot2^n$

Prove that for $n \ge 4$ $${{2n}\choose{n}} \ge n\times2^n$$ I tried like that: $T_4$: ${{8}\choose{4}} = 70 \ge 4\times2^4$ = 64 so it's ok $T_{n+1}$: $$\frac{(2n+2)!}{(n+1)!)(n+1)!} \ge ...
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0answers
43 views

Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$. Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I ...
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1answer
90 views

How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $

Can we find $$ \sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion? ...
2
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2answers
121 views

Sums of central binomial coefficients

Are there closed forms for $$\sum^n_{i=0} \binom{2i}{i}$$ and $$\sum^n_{i=0} \binom{2i}{i}^2$$? Also, how can these sums be approximated?
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1answer
376 views

binomial distribution(overbooking plane tickets)

I am having trouble with binomial distribution and this problem: an airplane has 200 seats, but 202 tickets are sold. Assume passengers do not show up with a probability of .03 independently. What is ...
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2answers
87 views

How can I compute this sum of binomial

Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.
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54 views

Close formula for triple sum binomial coefficient

I need to compute the following sum or to find a lower and upper bound that limit the sum: $\sum_{l=\frac{n+1}{2}}^n \binom{n}{l} \sum_{t=0}^{n-l} \binom{l}{t} 2^{l-t} \sum_{m=t}^{n-l} \binom{n-l}{m} ...
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2answers
103 views

Combinatorics identity algebraic proof

Prove that: $$\sum _{k=1}^nk\binom{n}{k}^2=n\binom {2n-1}{n-1}$$ I tried to prove it using induction: For n+1: $$ \begin{align*} \sum \:_{k=1}^{n+1}k\binom{n+1}{k}^2 &= \sum ...
1
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1answer
54 views

proving and Identity combinatorially

prove the Identity: (n-k)$\binom nk$ = n$\binom {n-1}k$ I have proven it algebraically but now I need to prove it combinatorially ( count something in two ways). Here is my attempt: theorem: ...
3
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1answer
73 views

Sum involving binomial coefficient

I want to analyse the following expression for $x \geq 0$: $$ \sum_{k = 0}^n (-1)^{k+n} \binom{n+k}{2k} x^k $$ I expect and want to prove that for $x \geq 4$, the expression tends to infinity as $n ...