0
votes
2answers
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.
3
votes
0answers
60 views

Generating functions to solve number of integer solution problem

If I have $x_1 + x_2 + x_3 =10$ with $1\leq x_1 \leq 5, \; 2 \leq x_2 \leq 6, \;3 \leq x_3 \leq 9$ I know that I compute $(t^1+\dots + t^5)(t^2 +\dots + t^6)(t^3+\dots +t^9)$ and look at the ...
2
votes
2answers
84 views

Alternating sum of binomial coefficients is equal to zero [duplicate]

Prove without using induction that the following formula:$$\sum_{k=0}^n (-1)^k\binom{n}{k}=0$$ is valid for every $n\ge1$. Progress For each odd $n$ we can use the ...
-1
votes
2answers
58 views

Squared binomial paradox?

When you square this $$(5-2)^2$$ you will get 49 $$ 5^2 - 2 * 5 * (-2) + (-2)^2$$ $$25 + 20 + 4 = 49$$ but if you do it like this (5-2) * (5-2) you will get 9 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
0
votes
0answers
45 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
-2
votes
1answer
66 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)!} \\ ...
5
votes
1answer
90 views

An inverse binomial summation.

I am looking for a closed form for this summation: $$ \sum_{j=1}^m\frac{r^{-j}}{j{m\choose j}} = \sum_{j=1}^m\frac{r^{-j}}{m{m-1\choose j-1}} = \frac1{rm} \sum_{k=0}^{m-1}\frac{r^{-k}}{{m-1\choose k}} ...
5
votes
3answers
698 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
4
votes
3answers
143 views

Taking Limits with Binomial Coefficients

I am interested in taking the following limit: \begin{equation} \lim_{n \to \infty}\frac{{n/2 \choose m}}{n \choose m}. \end{equation} Provided that $m$ is fixed the solution is: \begin{equation} ...
2
votes
5answers
61 views

How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$?

How to find the coefficient of $x^8$ in $(x-1) (x-2) . . .(x-10)$. Is there any general formula to solve this kind of problems?
3
votes
2answers
99 views

Simplify a triple sum

I need to find a closed form for this summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m}\choose{k}}}{j{m\choose j}}r^{k-j+i}$$ I posted this a long time ago, but today I found out ...
5
votes
2answers
111 views

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
1
vote
1answer
43 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 ...
5
votes
7answers
315 views

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

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
0
votes
0answers
38 views

Can this binomial coefficient term be simplified?

Can this be simplified? $$\binom{n}{k}\binom{k}{j}2^{-k}$$ assuming $k \le n$ and $j \le k$? I've tried expanding it in to factorials, but other than a $k!$ term, nothing seems obvious. ...
0
votes
2answers
330 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 ...
3
votes
2answers
77 views

$2^n=C_0+C_1+\dots+C_n$

Could anyone give me hints for this one? $2^n=C_0+C_1+\dots+C_n$ Thats all I can say and I know tricks like integrating or differentiating both side and then put $x=1$ or what ever we need. but ...
2
votes
2answers
212 views

Use induction and Newton's binomial formula to show that $\binom{n}{0}+\binom{n}{1}+\cdot+\binom{n}{n}=2^n, \forall n\in \mathbb N$ [duplicate]

Use induction and Newton's binomial formula to show that: $ i)$ $ \binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n, \forall n\in \mathbb N$ $ ii)$ ...
2
votes
0answers
281 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
2
votes
1answer
60 views

Proving Bernoulli's Inequality for $h<0$

I'm answering question 19 of chapter two of Spivak's Calculus and I can't seem to think of a way of doing it. I don't want to look up the answer so I thought I'd ask for a hint as to the general ...
0
votes
2answers
32 views

Alternative ways to write $ k \binom{n}{k} $

I am looking for a way to simplify $ k \binom{n}{k} $. I don't understand what effect the factor $k$ has on the formula. So can anyone please explain what $\ k\binom{n}{k} $ would equate to?
0
votes
2answers
91 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.
0
votes
0answers
75 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} ...
2
votes
1answer
77 views

Sum of series ${n\choose 2a}{a\choose 0}+ {n\choose {2a+2}}{{a+1}\choose 1} + {n\choose {2a+4}}{{a+2}\choose 2} + \ldots$

I wanted to check the rationality of the cosine function for some rational multiples of $\pi$. And I found out that, $\cos(n \cdot\arccos x)$ generates a polynomial in $x$ whose co-efficients have the ...
2
votes
1answer
42 views

Alternating binomial sum with intervals of two

Fix integer $n\geq 1$. Consider the number $$1-\binom{n}{2}+\binom{n}{4}-\binom{n}{6}+\cdots$$where the sum continues as long as the lower number in the binomial is $\leq n$. Is there a way to ...
1
vote
1answer
66 views

Simplification of Binomial Expansion.

How $$(x+h)^n-x^n=nhx^{(n-1)}\text{ ?}$$ My attempt : $$ \begin{align} (x+h)^n-x^n & =nhx^{(n-1)} \\[8pt] & =\left[\sum_{k=0}^{n}\binom{n}{k}x^{(n-k)}h^k\right]-x^n \\[8pt] & = ...
1
vote
1answer
51 views

Simplify Mathematical Expression

Can someone help me to simplify the following expression? I can assume b is small and $0<b<1$. $(C^{N}_{i})$ is the binomial coefficient. $$A=[(1-b)^{N} + \sum^{N}_{i=1} (C^{N}_{i} - 2 ...
-2
votes
1answer
102 views

Is this just a version of the binomial theorem?

I asked a question related to it and found something interesting (at least that is what I think)... Here is the link to the original question: What is the pattern of this sequence? I went through a ...
3
votes
3answers
101 views

Find the coefficient using binomial theorem.

What is the coefficient of $x^{20}$ in the expression: $$(x+1)^{10}.(x^2 -1)^8$$
5
votes
3answers
277 views

Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
4
votes
1answer
81 views

Binomial theorem for prime exponent

Could you explain to me why for prime $p$ we have the following? $$(x+y)^p - (x^p + y^p)= x^p + \binom{p}{1}x^{p-1}y + \binom{p}{2}x^{p-2}y^2 + \binom{p}{p-1}xy^{p-1} + y^p.$$ I found it here: ...
-2
votes
2answers
122 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 $$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$
5
votes
4answers
203 views

Proving $\binom{2n}{n}\le 4^n$ for all $n$ by smallest counterexample

Prove $$\binom{2n}{n}\le 4^n$$ for all natural numbers $n$ by smallest (minimal) counterexample. My attempt: First, $$\binom{2n}n = \frac{(2n)!}{(n!)^2} \le 4^n\;.$$ We know that $x\ne 0$ because ...
6
votes
3answers
271 views

How many solutions are possible to this equation?

Given $$A+2B+3C=N $$ where $N$ is a given positive integer. $A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$. How many solutions will be there to this equation?
7
votes
1answer
182 views

Closed-form expression for $\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}$?

Wolframalpha tells me that $$\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}=0$$ for $k>2$ However I have not been able to come up with a proof and I simply don't see how to do it. Does anyone ...
2
votes
3answers
210 views

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Can this identity be derived from the binomial theorem? $$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$$ I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it ...
0
votes
1answer
71 views

How did my textbook conclude this proof?

http://i.imgur.com/goUlA.png At the very last step, highlighted in red, it states that if this, then that sort of, and I'm not sure how those comparisons explain the issue of proving Pascal's ...
0
votes
1answer
101 views

Could anyone explain how my textbook did this simplification?

The book is talking about proving Pascal's triangle increases until the middle, until which point it decreases. Theorem 4.2 refers to the Multiplicative formula I just don't understand how that ...
0
votes
2answers
1k views

Calculate reciprocal square root via binomial expansion

I have to use a binomial expansion to evaluate $1/\sqrt{4.2}$ to $5$ decimal places. The answer from a calculator is $0.48795$ but I get $0.48202$, so I'm doing something wrong. I've also checked my ...
2
votes
2answers
178 views

Simplify $\sum_{i=0}^{n-1} { {2n}\choose{i}}\cdot x^i$

I am trying to simplify an expression involving summation as follows: $$\sum_{i=0}^{n-1} { {2n}\choose{i}}\cdot x^i$$ where $n$ is an integer, and $x$ is a positive real number. At a first ...
2
votes
2answers
3k views

Find the coefficient of $x^3y^2z^3$ in the expansion $(2x+3y-4z+w)^9$

The exercise says: In the expansion $(2x+3y-4z+w)^9$, find the coefficient of $x^3y^2z^3$. The formula to find the coefficient of $x_1^{r_1}x_2^{r^2}\dots x_k^{r_k}$ in $(x_1+x_2+\dots+x_k)^n$ ...
2
votes
2answers
320 views

Does this qualify as a proof? (Spivak's 'Calculus')

I'm working through Spivak's 'Calculus' at the moment, and a question about series confused me a bit. I think I have the solution, but I'm not sure if my "proof" holds. The question is: Prove ...
4
votes
0answers
144 views

How to transform series of series into series

I need to prove this equation. $$ \sum_{k=0}^{i-2} \left( e \space α(k+1)\space\frac{(-1)^{i+k+2}}{(i-k-2)!} \right) = \sum_{k=0}^{i-2} \frac{(i-k)^k}{k!} \space e^{i-k} (-1)^k\space where,\space α(i) ...
2
votes
2answers
211 views

simplify $(a_1 + a_2 +a_3+… +a_n)^m$

How to simplify this best $(a_1 + a_2 +a_3+... +a_n)^m$ for $m=n, m<n, m>n$ I could only get $\sum_{i=0}^{m}\binom{m}{i}a_i^i\sum_{j=0}^{m-i}\binom{m-i}{j}a_j ... $
3
votes
2answers
135 views

Why is the binomial coefficient related to the binomial theorem?

The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of ...
7
votes
4answers
339 views

Showing that $\lceil (\sqrt{3} + 1)^{2n} \rceil$ is divisible by $2^{n+1}$.

I have a question which has fluxommed me and my pals for the past few days. Any help or solution is welcome Show using Binomial theorem that the integer just after $(3^{1/2} + 1)^{2n}$ is divisble ...
4
votes
5answers
215 views

The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$

I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one: In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the coefficients of $x^3$?
7
votes
3answers
449 views

Hard elementary combinatorics problem

How does one compute (without brute force) the smallest integer $n$ such that $\binom{2n}{1}(-3)^0 + \binom{2n}{3}(-3)^1 + \binom{2n}{5}(-3)^2 + \cdots + \binom{2n}{2n-1}(-3)^{(n-1)} = 0$?
3
votes
5answers
496 views

Binomial theorem application

I have a question about the bonomial theorem, and in specifically, a question that I want help on. I have worked out the answer, but by manually expanding each and every alternative. However, I ...
1
vote
3answers
2k views

Calculate the expansion of $(x+y+z)^n$

The question that I have to solve is an answer on the question "How many terms are in the expansion?". Depending on how you define "term" you can become two different formulas to calculate the terms ...