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 binomial theorem is used in IP distribution?

I have read the binomial theorem/ Pascal triangle that they can be useful for IP ( Internet Protocol) address distribution . However, I am unable to understand it. How it can be applied for IP address ...
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2answers
104 views

Estimating sum with binomial coefficients

Lately when I was estimating complexity of some algorithm I came across this sum: $$\sum_{k=0}^n \binom {n}{k} \binom {n-k}{k}$$ Is it possible to find a closed-form expression for this sum, or at ...
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37 views

How to add binomial coefficients?

How do I add let say $\binom{n-2}{k-2} + \binom{n-2}{k-1}$ What are the steps of adding these together without breaking them down in factorials and add them up?
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Using Newton's binomial theorem to prove that a sum evaluates to $36^n-26^n$ [duplicate]

Using Newton's binomial theorem to argue that: $n \ge 1$ $$36^n - 26^n = \sum_{k=1}^{n}\binom{n}{k}10^k \cdot 26^{n-k}$$ my argument $$(26+10)^n = ...
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1answer
22 views

Use Leibniz' formula to show that the $(2n)$th derivative of $(2x^2 + 3x +1)sinx$ is $(-1)^n(2x^2+3x-8n^2+4n+1)sinx+(-1)^{n+1}(8nx+6n)cosx$ wrt $x$

If I let $f=f(x)=sinx$ and $g=g(x)=2x^2+3x+1$ and $D=$ First derivative wrt $x$, $D^2=$ Second derivative wrt $x$ and $D^n=$ $nth$ derivative wrt $x$ then, Leibniz' formula states that $\displaystyle ...
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3answers
92 views

What is wrong with this calculation of $\binom{\frac{1}{2}}{k}$?

The reason I ask this question is that I want to show that: \begin{equation*} \binom{\frac{1}{2}}{k} = -2\frac{1}{k}\binom{2k-2}{k-1}\left(-\frac{1}{4} \right)^k \end{equation*} \begin{align*} ...
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49 views

Yet another sum involving binomial coefficients.

Given $A,B,N \in \mathbb N$ Is there a closed form for this expression? $$\sum_{n=1}^N n \binom{A}n \binom{B}{N-n} $$ If there is such, can you give a proof? EDIT: $A,B \geq N$
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195 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
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18 views

Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
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2answers
32 views

Understanding a combinatorial relation.

I would like some insight as to why the following expression is true. $$\sum_{i=0}^n {{n}\choose{i}} 2^{n-i} = 3^n $$ I arrived at this relation in solving a subset problem, and I understand the ...
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3answers
35 views

What does this symbol mean in this equation?

I am reading up on n-choose-k problems (Binomial Coefficients). Wikipedia gives a multiplicative solution that is more efficient: I've taken a few calculus courses, and it reminds me of how you ...
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0answers
33 views

All or no Heads from biased coin tosses

What is the probability that all five tosses of a biased coin with $P(H)=0.28$ are (a) Heads and (b) Tails? (c) What is the probability of at least one Head? (a) Heads $Pr(5\ Heads) = {5 \choose 5} ...
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1answer
15 views

Prove identity based on binomial theorem

$\displaystyle \sum_{r=0}^{n-1} {2n-1 \choose r} = 2^{2n-2} $ Perhaps it can be proved by using sum of all combinations from r=0 to r=n is 2 to the power of n.
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2answers
26 views

Alternating sum of a part of a row of Pascal's triangle

$\displaystyle \sum_{r=0}^m(-1)^r{n \choose r}=(-1)^m{n-1 \choose m}$ if $m$ is less than $n$. This question actually consists of two part that is when $m$ is less than $n$ and when $m$ is equal to ...
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79 views

Solving equation involving binomial function

Solve for $x$ in terms of $i$ and $j$: $$ \binom{x}{i} = j $$ where $x$ is Real; $i$ and $j$ are Integers: $x \geqslant i$, $i \geqslant1$, $j \geqslant 0$. I came across this problem while trying ...
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1answer
61 views

Conditions for equality of two binomial sums

Let $k,r,n$ be integers such that $0<k,r<n$. Let $$K=\sum^n_{i=k}k^{n-i}\binom{n-k}{i-k}^2k!(i-k)! \,\text{ and }\, R=\sum^n_{i=r}r^{n-i}\binom{n-r}{i-r}^2r!(i-r)!.$$ How to show that ...
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1answer
26 views

Prove that $F(N/2;N,x)+F(N/2;N,1-x)=1$ where $F$ is binomial CDF

I have the following claim: $$F(N/2;N,p)+F(N/2;N,1-p)=1$$ where $F$ is a binomial CDF with exactly $N/2$ successes in $N$ total trials, and with each trial having success probability $p$. Is it ...
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1answer
19 views

How to articulate this expression

I was walking a student through the binomial expansion process and remarked that I prefer Pascal's triangle to generate the coefficients. He also needed to know this way of producing the numbers. ...
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2answers
90 views

Binomial theorem / Generating function

I had trouble figuring out if the following equality holds by applying the binomial theorem and using generating functions. Could anyone please shed some light? Any help is greatly appreciated. $${n ...
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1answer
26 views

Variance in offspring genotypes. Binomial distribution

Background Here is first some vocabulary: Diploid: phase in the life cycle where the individuals carry two chromosomes of each type, just like in humans (exception of the sexual chromosomes). ...
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2answers
50 views

Problem proving $ \sum_{r=0}^{n-1} \binom{2n-1}{r} = 2^{2n-2} $

I'm stuck at proving the following. $$ \sum_{r=0}^{n-1} \binom{2n-1}{r} = 2^{2n-2} $$ I know that I have to use the Binomial theorem like this, letting x=1,y=1 in $(x+y)^{2n-1}$ $$ ...
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1answer
52 views

Prove $\sum_{i=0}^{i=x} {x \choose i} {y+i \choose x}+\sum_{i=0}^{i=x} {x \choose i} {y+1+i \choose x}$

How to prove that $$\sum_{i=0}^{i=x} {x \choose i} {y+i \choose x}+\sum_{i=0}^{i=x} {x \choose i} {y+1+i \choose x}=\sum_{i=0}^{i=x+1} {x+1 \choose i} {y+i \choose x}$$ ? I tried to break the right ...
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3answers
33 views

Proving $\sum_i \binom{k}{k-i} \binom{n-k}{i} = \binom{n}{k}$

I am in the middle of a probability question. The question is indeed simple. For the sake of clarity of the notation, I also include the question here, which is from Sheldon M. Ross's Introduction to ...
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2answers
98 views

Sums of binomial coefficients

Does anyone know something about the following sums? $$ S_m(n)=\sum\limits_{k=o}^n(-1)^k{mn\choose mk} $$ Notice that $S_m(n)=0$ for odd $n$, so we only consider $S_m(2n)$. It holds that $S_0(2n)=1$, ...
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1answer
130 views

Placing m books on n shelves

If we let m and n be integers with $m \ge n \ge 1$. how many ways are there to place m books on n shelves, if there must be at least one book on each shelf? the order matter. How do I solve this, do I ...
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43 views

Solve Identity about Combination

Find the values of a and b such that $\binom{2n}{2} = a\binom{n}{2} + b(n^2)$ This is a past year question about Introduction of Combinatorics in my university.
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35 views

How many ways can the school choose a President Vice President?

There are n >= 4 students. The school has a Board of Directors, consisting of one president and three vice-presidents. The entire board consists of four distinct students. How can I prove that ...
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4answers
100 views

Question about ${n+1\choose k} = {n\choose k} + {n\choose k-1}$ proof?

I've found past proofs of this problem and for the most part I'm able to follow. $$\eqalign{{n\choose k}+{n\choose k-1}&= {n!\over (n-k)!k!}+ {n!\over (n-(k-1))! (k-1)!} \text{ (step 1)}\cr ...
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2answers
59 views

The sum of binomial coefficients up to $k\le n/3$ does not exceed the $k$th coefficient

How would you prove the following (for when $k\leq\frac{n}{3}$)? $$\sum_{i=0}^{k-1} \binom ni \le \binom nk$$
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1answer
200 views

A urine test, the VMA test

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about 70 % of cases of neuroblastoma. It has been proposed ...
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1answer
31 views

Binomial coefficients and order of infinity

Which among $$ \left(2\,k+1 \atop j\right),~~j=1,3,5,...,2\,k+1 $$ has the larger order of infinity when $k\rightarrow\infty$? I am pretty sure that the largest order is reached around $j=k$ but I ...
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1answer
44 views

What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$?

What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$? So, would I start by using the binomial or multinomial theorem? Not entirely sure where to start here?
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29 views

What is the coefficient of$ x^6y^1$ in the expansion of $(3x^2+y)^4$?

So this is what I have so far. $(3x^2)^4$ + $\binom{4}{1}(3x^2)^3(y)$ Why is the answer not 4? How do I continue?
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34 views

A drug treatment [closed]

A certain drug treatment cures 90 % of cases of hookworm in children. Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from ...
3
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1answer
27 views

Show by committee selection argument

First post in Stack Exchange and feel bad to be in need of help. But, I'm having a hard time understanding this one or rather showing the argument. $\binom{n}{k} = \binom{n-2}{k-2} + ...
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50 views

What is the combinatorial interpretation of the product of binomial coefficients?

Full disclosure: This question is relating to a homework question. It's not a homework question itself, but rather a clarifying question to help myself get a handle on the actual question. Suppose I ...
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5answers
125 views

Calculate the binomial sum $ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i} $

I need any hint with calculating of the sum $$ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i}. $$ Maple give the strange unsimplified result $$ I_n={\frac {1/12\,i\sqrt {3} \left( - \left( \left( ...
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1answer
34 views

Sum of products of binomial coefficients is equal to another binomial coefficient [duplicate]

Need help in proving (by induction or by combinatorics) the following statement Is it possible to do it by induction? there are 3 veriables and I think I cannot easily do it by induction. Correct? ...
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1answer
90 views

Combinatorial Analysis: Fermat's Combinatorial Identity

I was looking through practice questions and need some guidance/assistance in Fermat's combinatorial identity. I read through this on the stack exchange, but the question was modified in the latest ...
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2answers
52 views

Sum of Binomial Coefficient products

I am trying to prove that $$\sum\limits_{y=0}^d \frac{{2x \choose y} {2d-2x \choose d-y} }{2d \choose d} = x $$ So far, I have tried using induction on $d$ but I am having trouble using the ...
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1answer
51 views

Prove combination identity

$ \sum_{k=0}^n {2k \choose k} {2n-2k \choose n-k} = 4^n $ I tried with mathematical induction only to fail. Is this formula related to some special function like Beta, Gamma, etc?
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33 views

Probability for a 'pair' to occur when rolling 5 dice

5 fair dice are rolled. A pair is defined to be any number that shows up twice, while the rest of the dice show different numbers (to the number on the pair and to each other). I am looking for the ...
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2answers
46 views

Gosper's Identity $\sum_{k=0}^n{n+k\choose k}[x^{n+1}(1-x)^k+(1-x)^{n+1}x^k]=1 $

The page on Binomial Sums in Wolfram Mathworld http://mathworld.wolfram.com/BinomialSums.html (Equation 69) gives this neat-looking identity due to Gosper (1972): $$\sum_{k=0}^n{n+k\choose ...
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1answer
31 views

binominal inequality - checking

I have to show that $\displaystyle \binom{n}{k}<\binom{n}{l}$ for $\displaystyle 0 \le k <l \le\frac{n}{2}$ where $n,k,l$ are an integers. I think I solved it but I'm not sure if my approach is ...
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4answers
162 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
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1answer
64 views

Proving that $\sum_{i = 0}^{m}{\binom{k+i}{i} \binom{n-i}{m-i}} = \binom{n+k+1}{m}$

Goal: prove that $\displaystyle\sum_{i = 0}^{m}{\binom{k+i}{i} \binom{n-i}{m-i}} = \binom{n+k+1}{m}$ How to give a combinatorial argument, i.e. counting in two ways, for this problem? I tried by ...
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21 views

Transformation of a sum

I want to prove the following or a similar result: For $1\le k \le n$ \begin{align}&1-\sum\limits_{j=k+1}^n\binom nj(1-x)^jx^{n-j}~~~~~~(1)\\ ...
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1answer
26 views

Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.

What function satisfies the following: Let the matrix: $$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&0 \\ ...
4
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4answers
100 views

Solve for $n$, where $n$ is a positive integer

I have $$ {n \choose 2} = 21 $$ and as the title mentions I have to solve for $n$, but so far all I have managed to get to is $$n^2 -n =42 $$ and from there I'm completely lost. Any hints would ...
2
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3answers
63 views

Show that $ \sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1} $?

I can't resolve this exercise and I need a tip. $$ \sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1} $$ where $ n \geq s $.