3
votes
2answers
40 views

Binomial dependent on a Poisson

I have been working on a problem with a binomial rv dependent on a poisson rv and have worked through to this point: $P(X=x) = \sum_{n=x}^{\infty} \dfrac{n!}{x!(n-x)!} p^x(1−p)^{n−x} ...
0
votes
1answer
24 views

Finding $V(X)$ when you don't have a density/distribution function.

I just did the first part of this problem: You have a lot of $50$ items and are taking a sample size of $15$. In the lot $3$ items are defective. The lot is accepted if the number of defective items, ...
2
votes
1answer
26 views

Binomial thereom to figure out coefficents

Use the binomial theorem to find the coefficient of $x^8y^5$ in $(x + y)^{15}$ My textbook shows how to do this looking at the coefficents of Pascal's triangle but, I know theres another way using ...
-2
votes
1answer
56 views

Simplifying the sum of two consecutive binomial coefficients

I've been trying to figure out how they simplified the right side of the equation all the way down. $$\eqalign{ \binom nk + \binom n{k-1} & = \dfrac{n!}{(k)!(n-k)!}+\dfrac{n!}{(k-1)!(n-k+1)!} \\ ...
3
votes
4answers
111 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
4
votes
1answer
63 views

Extracting a coefficient from a generating function

Background: I am working on an exercise relating to Skolem $k-$subsets with index $k$ in Goulden and Jackson's Combinatorial Enumeration text and they broke it down to finding the coefficient of $x^n$ ...
2
votes
1answer
44 views

Number of nodes with an even number of children in an ordered tree (AKA Plane Planted Tree)

I am looking for verification for my attempt at the solution. I have found that my answer disagrees with an answer I found here: Extract Coefficients From A Function Problem at hand: For a plane ...
10
votes
1answer
204 views

Prove that $\exp(\log(\frac{1}{1-x})) = \frac{1}{1-x}$

I am trying to prove this directly by comparing the coefficients in the two series rather than using formal calculus. Here is what I have so far, but I think I made a mistake: \begin{align*} ...
0
votes
1answer
75 views

Binomial Expression

Please give me feedback on my answer to this question. Question: For all $ n\geq1:\binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n} $ is equal to $ ...
0
votes
2answers
69 views

Evaluate the sum $\sum_{0\leq j < k\leq n}\binom{n}{j}\binom{n}{k}$

Could someone give me a hint on how to do this? I believe I know what the answer to be (I computed some low values and checked on OEIS). However, I was hoping someone would be able to explain to me ...
0
votes
2answers
21 views

Question about Binomial Distribution

The chance of a rose flower blooming is .28. You are going to plant 5 rose flowers, what are the chances of 4 of them blooming? I was thinking the answer would be 35% since 28%x5=140 and 140/4=35. ...
0
votes
0answers
22 views

Binomial expansion, the greatest term…

My question is related to binomial expansion, and more precisely the greatest term in expansion. Is it right that the formula for finding the greatest term is $$T_k\ge T_{k+1}$$? Now going to the ...
4
votes
1answer
189 views

Binomial Summation

The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then ...
6
votes
1answer
94 views

Help with a Binomial Identity: $\sum_{k=0}^{\infty} \binom{n+k}{k}2^{-k} = 2^{n+1}$

The following is a problem from the 5th edition of Niven's An Introduction to the Theory of Numbers: Problem 23 of Section 1.4 asks us to prove that $$\sum_{k=0}^{\infty} \binom{n+k}{k}2^{-k} = ...
1
vote
2answers
102 views

Binomial Expansion.

So I had a question: Prove that for $n \geq 1$, $${n \choose 1} + 2{n \choose 2} + 3{n \choose 3} + ...+ n{n \choose n} = n2^{n-1}$$ So my idea was to take the binomial expansion of $(1+1)^n$ which ...
1
vote
1answer
107 views

Binomial Distribution - independence

I have the following problem that I'm stuck on a few parts. ...
0
votes
1answer
30 views

Binomials for getting probability of standard deviation

I have the following problem which I am stuck on the second part. Suppose that $30\%$ of all students who have to buy a text for a particular course want a new copy whereas the other $70\%$ want a ...
0
votes
1answer
36 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
0
votes
5answers
225 views

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
5
votes
2answers
574 views

Proving $\sum_{k=1}^n{2k-1\choose k}{2n-2k+1\choose n-k+1}=4^n-{2n+1\choose n+1}$

Some background. I was asked to find an arithmetic function $f$ such that $f*f=\mathbf 1$ where $\mathbf 1$ is the constant function 1 and $*$ denotes Dirichlet convolution. I was able to prove that ...
2
votes
0answers
86 views

How to prove this combinatorial identity

I am wondering how to prove the following identity: $$\sum_{k=0}^r {r-k \choose m} {s \choose k-t} (-1)^{k-t} = {r-t-s \choose r-t-m}$$ It seems that I can negating the upper index of ${s \choose k-t} ...
1
vote
1answer
41 views

Problem with raising parentheses to powers

Simply math question, lets say I have $(2x^2)^3$.Is this equal to $8x^6 , 2x^5, 2x^6$, or $8x^5$ ? It is a simple problem but what confuses me is do if I multiply the coefficient separately from the ...
0
votes
2answers
280 views

Binomial Theorem Practical Problem

I have been studying the 'Binomial Series', Chapter 16, Pg.125 within the Engineering Mathematics Book by John Bird. After completing this section I have attempted to complete the exercises for ...
0
votes
2answers
66 views

Binomial Theorem question

Find the coefficient of $x$ in the expansion of $$\left(1-2x^3+3x^5\right)\left(1+\frac{1}{x}\right)^5.$$ Answer is $154$, but how?
3
votes
1answer
189 views

Find the sum of the series.

I need to find the following sum: $$\sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s}$$ First I tried to simplify this: $$\begin{split} \sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s} &= ...
3
votes
3answers
164 views

Find the 100th derivative of $x \sinh(2x)$

If $f(x) = x \sinh(2x)$, find $f^{({100})}(x)$. My (Incorrect) working so far: Using Leibniz' Formula for derivatives: $$(fg)^{(n)}=\sum_{k=0}^n{n\choose k}f^{(k)}g^{(n-k)}$$ ...
6
votes
3answers
96 views

Show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$

In this question we are asked to show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$ What I did: $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \sum_{k=2012}^{n} 2^k*1^{n-k}\binom{n}{k} \leq ...
1
vote
4answers
66 views

Binominal Theorem

Could anyone help me with homework or give me a hint? Any help would be highly appreciated. Given a set of N distinct objects: How many ways are there to pick any number of them to be in a pile ...
2
votes
1answer
37 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
votes
2answers
85 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 ...
1
vote
1answer
47 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 ...
3
votes
1answer
185 views

Proving combinatorial identity

I need to prove following combinatorial identities: $$ \sum\limits_s(-1)^s\binom{p+s-1}{s}\binom{2m+2p+s}{2m+1-s}2^s=0 $$ $$ ...
1
vote
0answers
54 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 ...
3
votes
1answer
107 views

Find the constant $c$ in the equation $\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.$

Find the constant $c$ in the equation $$\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.$$ I've tried to use this asymptotics $$C_n^k \sim \frac{n^m}{m!} \sim e^{m\ln n ...
5
votes
1answer
121 views

Find asymptotic for $s(n)=\min\{m\in{\mathbb N}\mid C_n^m\cdot e^{-m^3/(\ln m)^{10}}<1\}$

I have some strange function: $s(n)=\min\{m\in {\mathbb N} \mid C_n^m\cdot e^{-m^3/(\ln m)^{10}}<1\}$ and I need to find asymptotics for it. I have a solution for this except one last step, I ...
1
vote
5answers
77 views

Calculate $\lim_{n\to\infty}\binom{2n}{n}2^{-n}$

I would like to show that: $$\lim_{n\to\infty}\binom{2n}{n}2^{-n} = \infty$$ I have gotten as far as: $$ \binom{2n}{n}={(2n)!\over (n!)^2}=({n\over1}+1)({n\over2}+1)(\dots)({n\over n}+1)\ge2^n $$ But ...
0
votes
1answer
65 views

Inequality with a sum and factorial

For a homework assignment we have the following question that I'm stuck on. Let $ 0 \leq y \leq 1 $ be given. $\forall m \in \mathbb{N}$, define $ \displaystyle S_m(y)=\sum_{k=0}^m \binom{m}{k}y^k$. ...
2
votes
3answers
412 views

Fermat's Combinatorial Identity: How to prove combinatorially?

$$\binom{r}{r} + \binom{r+1}{r} + \binom{r+2}{r} + \dotsb + \binom{n}{r} = \binom{n+1}{r+1}$$ I don't have much experience with combinatorial proofs, so I'm grateful for all the hints. (Presumptive) ...
0
votes
3answers
905 views

Find a constant in a binomial expansion

Find the constant 'a' in the binomial expansion: $(1-2x)(1+ax)^{10}$ given that the coefficient of $x^6$ is $0$. I get 9.86, is this correct?
1
vote
2answers
66 views

Binomial Expansion involving two terms?

How would you find the 4th term in the expansion $(1+2x)^2 (1-6x)^{15}$? Is there a simple way to do so? Any help would be appreciated
1
vote
4answers
456 views

Binomial coefficient question?

I'm unsure how to do these types of questions, so any help would be great: Find the coefficient of $x^2$ in the expansion of $(x+1/x)^3(x-1/x)^5$ Thanks
0
votes
2answers
56 views

Prove that $\sum_{k=0}^{n}\frac{\binom{n}{k}}{k+1}=\frac{2^{n+1}-1}{n+1}$

I want to prove the following $$\sum_{k=0}^{n}\frac{\binom{n}{k}}{k+1}=\frac{2^{n+1}-1}{n+1}$$ I need some hint how to start. thanks!
3
votes
2answers
93 views

Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$ [duplicate]

I want to prove the following: $$\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$$ what I did is(use binominal): $$\binom{n}{k}X^k\cdot 1^{n-k} = (X+1)^n$$ $$k\binom{n}{k}X^k\cdot 1^{n-k} = k(X+1)^k-1$$ now ...
1
vote
1answer
49 views

Find the coefficients($x^{15}$,$x^{11}$) of the function $(2x^2-\frac{3}{x})^{10}$

I want to find the coefficients ($x^{15}$,$x^{11}$) of the function $(2x^2-\frac{3}{x})^{10}$ I would like to get some advice how to continue thanks!
2
votes
0answers
99 views

Proving an equality involving binomial coefficients and summations

Question: $$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right ...
1
vote
2answers
39 views

How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$

So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do ...
1
vote
2answers
1k views

Binomial Expansion Word Problem (Creating a Equation)

I was working on my math textbook (Nelson Functions 11) and came across the following word problem. This question is shown in the "Pascal's Triangle and Binomial Expansions" section of the book. ...
4
votes
4answers
118 views

A binomial inequality with factorial fractions

Prove that $$\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$$ for $n>1 , n \in \mathbb{N}$.
1
vote
2answers
68 views

$\sum_{i=0}^n (-1)^k \cdot C(n,k)$

Use the Binomial Theorem to Show $$\sum_{i=0}^n (-1)^k \cdot C(n,k)$$ I'm not sure where to start here . . . I know it is missing an $a^{n-k}$ and a $b^{k}$ term that maybe I should set to be 1?
-2
votes
2answers
118 views

Calculation of binomial sum $\displaystyle \sum_{r=1}^{n}r.\binom{n}{r}x^r.(1-x)^{n-r} = \;\;?$ [closed]

How can I calculate $$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$