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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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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1answer
99 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 ...
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1answer
38 views

Showing equivalence of two binomial expressions

I wish to show that $\sum_{k=0}^n {n\choose k}(\alpha + k)^k (\beta + n - k)^{(n-k)} = \sum_{k=0}^n {n\choose k}(\gamma + k)^k (\delta + n - k)^{(n-k)}$ given that $\alpha + \beta = \gamma + \delta$. ...
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23 views

Choice problem: number of pairs can be formed out of a set with odd cardinality

There are 17 languages (at a meeting) and for every two languages there is one interpreter assigned. The number of pairs we can form out of 17 languages is $\binom{17}{2} = 136$. So 136 interpreters ...
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4answers
69 views

Prove that $\sum_{k=1}^nk^2{n\choose k}^2=n^2 \binom {2n-2}{n-1}$

Please help me / give a hand with combinational prove for: $$ 1^2 \binom n 1 ^2 + 2^2 \binom n 2 ^2 + \dots + n^2 \binom n n ^2 = n^2 \binom {2n-2}{n-1}$$
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2answers
48 views

Why does $ \frac{b^n-a^n}{b-a}=\sum_{k=1}^nb^{n-k}a^{k-1}$?

Trying to work through the answer in this question: The inequality $b^n - a^n < (b - a)nb^{n-1}$
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5answers
421 views

Choice Problem: choose 5 days in a month, consecutive days are forbidden

I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. Example 3.19. A medical student has to work in a hospital for five days in January. ...
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4answers
161 views

Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction

I have to prove that: $$ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $$ I don't want a complete solution, but only a hint.
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2answers
47 views

Sums with squares of binomial coefficients multiplied by a polynomial

It has long been known that \begin{align} \sum_{n=0}^{m} \binom{m}{n}^{2} = \binom{2m}{m}. \end{align} What is being asked here are the closed forms for the binomial series \begin{align} S_{1} &= ...
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33 views

difference between independent binomial variables

It is well known that if $X \sim B(m, p)$ and $Y \sim B(n, p)$ are independent then $X+Y \sim B(m+n, p)$ but what is the distribution of $X-Y$? Here is what I have tried. $\Pr[X-Y = c] = \sum_{i=0}^n ...
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3answers
51 views

Alternating sum with binomial coefficients

$\sum_{k=0}^{49}(-1)^k\binom{99}{2k}$ = ? I've tried expanding the binomial coefficient in its factorial form and can't seem to get to manipulate it in a way that solves the expression. ...
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2answers
34 views

proving counting problems?

Let n >= 1 be an integer. We consider passwords consisting of n characters, each character being a digit or a lowercase letter. A password must contain at least one digit. How do I show that the ...
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0answers
52 views

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
112 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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0answers
38 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
35 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
98 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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53 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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1answer
206 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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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
43 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
40 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
35 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
42 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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2answers
82 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
63 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
28 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
21 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
112 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
34 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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56 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
54 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
36 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
108 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
151 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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44 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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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
102 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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1answer
68 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
216 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
34 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
52 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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3answers
33 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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1answer
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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2answers
52 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
136 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( ...
0
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1answer
38 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? ...