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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3
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2answers
54 views

Formula for $\sum_{k=0}^n k^d {n \choose 2k}$

If $d \geq 1$ is an integer, is there a general formula for $$\sum_{k=0}^n k^d {n \choose 2k}\,?$$ We know that $\sum_{k=0}^n k {n \choose 2k} = \frac{n2^n}{8}$ and $\sum_{k=0}^n k^2 {n \choose 2k} = ...
1
vote
1answer
42 views

Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
1
vote
2answers
36 views

Is $n\binom{\epsilon n}{t}>t\binom{n}{t}$ for large $n$ and fixed $\epsilon$ and $t$

Let $\epsilon$ and $t$ be fixed numbers with $t$ and integer. I came across the following inequality in a counting problem. $$n\binom{\epsilon n}{t}>t\binom{n}{t}.$$ I want to show that for $n$ ...
3
votes
1answer
41 views

How prove $\binom{n}{m}\le\left(\frac{en}{m}\right)^m$ [duplicate]

Show that $$\binom{n}{m}\le\left(\dfrac{en}{m}\right)^m$$ where $0<m\le n,m,n\in N^{+}$ My idea: since ...
1
vote
1answer
39 views

Addition of Binomial Coefficients

$$\left[\binom n{k-1} + \binom nk\right] + \left[\binom nk + \binom n{k+1}\right] = \binom{n+1}k + \binom{n+1}{k+1}$$ Can anyone else explain to me, without using Pascal's triangle, how this ...
0
votes
1answer
23 views

Binomial coefficients in series

Here's a tricky one which I don't know how to start so any help would be appreciated. Show that no 4 consecutive binomial coefficients can be in AP and no 3 consecutive binomial coefficients can be in ...
4
votes
4answers
90 views

Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

An exercise in the first chapter of Discrete Mathematics, Elementary and Beyond asks for a proof of the following identity: $$ {n \choose 2} + {n+1 \choose 2} = n^2 $$ The algebraic solution is ...
0
votes
0answers
30 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
0
votes
1answer
21 views

Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
2
votes
3answers
140 views

Sum of binomial coefficients with three variables

What's the sum of coefficients of $(a+b+c)^8$? Thanks in advance!
1
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4answers
77 views

Maths binomial theorem question. [on hold]

Using the binomial theorem find the coefficient of $x^2$ in the expression $(1+x+x^2)^{10}$. Please help me by explaining how to proceed with this question.
0
votes
1answer
18 views

number of terms in the expansion containing powers of $x$

How do i find the number of terms containing powers of $x$ in the expansion of: $$(1+x)^{100}(1+x^2-x)^{101}$$ I tried using $(1+x)((1+x(1+(x)^2-x))^{100})$ which simplified into : ...
4
votes
2answers
42 views

Is there a simpler way to compute this sum?

For any given positive integers $m$, $n$ and $q$, such that $m\leq n$ the following sum $$S_p= \sum_{p=0}^m(-1)^{p+q} \binom mp \binom mq \binom np \binom nq\frac{p! q! (m+n-p-q)!}{m! n!}$$ is equal ...
0
votes
2answers
33 views

Integral equation involving Binomial distribution

I am trying to find the form of a function $u^{(n)}(p)$ which satisfies $\forall k \in [0,n] \int_0^1 dp\, u^{(n)}(p) \binom{n}{k} p^k(1-p)^{n-k} = \frac{1}{n+1}$. This is a private case of a more ...
0
votes
1answer
22 views

Count no. Of ways

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.
2
votes
2answers
42 views

limit of $2n$-th root of $2n$-th central binomial coefficient — $\lim_{n\to \infty}{2n \choose n}^{\frac{1}{2n}}$

$\lim_{n\to \infty}{2n \choose n}^{\frac{1}{2n}} = 2$ according to wolframalpha. Does anyone see how to get this limit of $2$? Is it a numerical estimate or analytically exact?
1
vote
1answer
30 views

Proof on Divisibility of Binomial Coefficients

Prove that $\exists \ i$ $(0 \lt i \lt n)$ such that $$ n \nmid {n \choose i} $$ $\forall \ n$ such that $n \gt 0$ and $n$ is a composite Number.
1
vote
2answers
36 views

Sum of Binomial Series of form $\binom{2000}{3k-1}$

Find the Value of $$ \binom{2000}{2}+\binom{2000}{5}+\binom{2000}{8}+\cdots+\binom{2000}{1997}+\binom{2000}{2000}$$
1
vote
2answers
39 views

Inequality with two binomial coefficients

I am having trouble seeing why $$ \binom{k}{2} + \binom{n - k}{2} \le \binom{1}{2} + \binom{n - 1}{2} = \binom{n - 1}{2} $$
2
votes
3answers
52 views

Set of solutions for a binomial inequality

I bumped into the following inequality: $${a-b\choose c}{a\choose c}^{-1} \le \exp\left(-\frac{bc}{a}\right)$$ Playing with it a little bit, trying to bound it asymptotically for large $a$'s, using ...
2
votes
1answer
22 views

Finding the value of an expression involving co-efficients in binomial expansion

Let $$(1+x)^n = C_0 +C_1.x+C_2.x^2 +: : :+C_n.x^n,$$ n being a positive integer. Then find the value of the following expression: $$(1+C_0/C_1)*(1+C_1/C_2)*.....*(1+C_{n-1}/C_n)$$
0
votes
1answer
31 views

Show that if $a,k\in \mathbb{Z}$ with $0\leq k \leq a$, then $\binom ak=\frac{a!}{k!(a-k)!}=\binom {a}{a-k}$.

I'm reading Ghorpade's A Course in Calculus and Analysis. Given $a\in \mathbb{R}$ and $k\in \mathbb{Z}$, the binomial coefficient associated with $a$ and $k$ is defined by: $$\binom ak = ...
3
votes
1answer
45 views

Very loose bound on sum of first binomials

Let $n\geq k\geq 2$. Is it always true that $$\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\leq n^k?$$ The left-hand side is dominated by the term $\dfrac{n^k}{k!}$, so the statement should be true. ...
0
votes
1answer
22 views

Cumulative Binomial Distribution function , Solve for n (trials)

how can one solve for $n$ in the Cumulative Binomial Distribution Function $P=\sum_{i=0}^{i=c-1} {n \choose i} p^{i}(1-p)^{n-i}$. thanks in advance, D.
1
vote
3answers
40 views

Sum of certain binomial coefficients

$$\sum_{k=0}^{m} \frac{(q+k)!}{k!q!}$$ I do not know how to even start this problem. Any general tips on these types of problems will also be welcomed.
4
votes
3answers
184 views

How to closed the sum $\displaystyle \sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$

How to closed the sum $\displaystyle S=\sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$ I'm trying divide two cases $n$ odd and $n$ even. I predict that ...
-2
votes
2answers
37 views

Find the coefficient of $x^9y^3$ in the expression of $(2x-3y)^{12}$ [closed]

How do I find the coefficient of $x^9y^3$ in the expression of $(2x-3y)^{12}$?
5
votes
3answers
665 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 ...
0
votes
0answers
19 views

closed form of a specific crazy summation?

How can I find the closed form of $f_2 + f_4 + ...+ f_{2m}$ where $\sum\limits_{m=1}^\infty f_{2m} = u_{2m-2}- u_{2m} $ where $u_{2m} = \binom{2m}{m} 2^{-(2m)}$ and $u_{2m-2} = \binom{2m-2}{m-1} ...
4
votes
0answers
76 views

How to prove this indentity $\binom{100}{0}^2-\binom{100}{1}^2+\binom{100}{2}^2-…-\binom{100}{99}^2+\binom{100}{100}^2=\binom{100}{50}$ [duplicate]

I don't know how to prove this identity: $\binom{100}{0}^2-\binom{100}{1}^2+\binom{100}{2}^2-\binom{100}{3}^2+...-\binom{100}{99}^2+\binom{100}{100}^2=\binom{100}{50}$
0
votes
1answer
44 views

closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$

how to get closed form for $$\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$$ I tried to write binominal in term of gamma function but I got no result what is your suggest to solve the problem ?
5
votes
2answers
192 views

What is $\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$

What is $$\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$$
2
votes
1answer
49 views

What is $\lim_{n\to\infty} \sum_{k=1}^{n/2} \left(\frac{en}{2k}\right)^{2k} \frac{1}{\sqrt{\pi k}}^{\frac{3n}{\log_2{n}}}\,?$

What is $$\lim_{n\to\infty} \sum_{k=1}^{n/2} \left(\frac{en}{2k}\right)^{2k} \frac{1}{\sqrt{\pi k}}^{\frac{3n}{\log_2{n}}}\,?$$ Numerically it seems to be $0$.
4
votes
3answers
69 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
2answers
57 views

Closed form for $\prod_{k=1}^n \binom{k^2+2k}{k^2}$

Does anybody know how I can find a closed form for the expression $$ \prod_{k=1}^n \binom{k^2+2k}{k^2}? $$ There are many ways to handle such things, but with sum instead of product. Here, I have no ...
-1
votes
2answers
42 views

Prove $k\binom nk=n \binom{k-1}{ n-1}$ algebraically.

I need to prove $k\binom nk=n \binom{k-1}{ n-1}$ where $n$ and $k$ are integers with $1\leq k\leq n$ using an algebraic proof. I solved the left side which is $\binom nk$ using the pascals identity ...
2
votes
1answer
76 views

Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$

Let $n,m$ are positive integers satisfy the condition $n\ge m>0$ Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$
5
votes
1answer
149 views

Limit of $ \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} 2^{-2nk} \binom{n}{2k}\left(\binom{2k}{k}^n\right)$

I can see numerically that $$\lim_{n \to \infty} \sum_{k=0}^{\lfloor n/2 \rfloor} 2^{-2nk} \binom{n}{2k}\left(\binom{2k}{k}^n\right) = 1$$ but how can you prove this? Using Stirling's approximation ...
1
vote
1answer
34 views

Simplifying Sum of Subsets

Given sets $A$ and $R$ such that $R \subseteq A$ and a number $x \leq |A|$, I am trying to simplify the following sum: $$\begin{equation*} \sum_{R \subseteq W \subseteq A : |W| = x} \Big( \sum_{Y ...
2
votes
1answer
25 views

Probability of two independent random variables being equal

Assume that $X$ and $Y$ are two independent random variables that follow the binomial distribution of parameters $p$ (the probability of one success) and $n$ (the number of trials). I was wondering ...
2
votes
5answers
59 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?
2
votes
0answers
32 views

Binomial coefficients inequality

It seems to me that there should be a simple way to prove that $$ \binom{n}{s+1+a} + \binom{n}{a} \leq \binom{n}{s} $$ For $s > n/2$ and $a < n-s$. However it looks like I'm missing it. Any ...
-1
votes
0answers
26 views

No. of odd and even numbers in binomial expansion

For a given number n, there would be n+1 terms in binomial expansion. Out of them how many will be odd valued?
2
votes
5answers
103 views

coefficient of $x^2$, in $(1+x+x^2)^{10}$

How to find coefficient of $x^2$, in $(1+x+x^2)^{10}$, without actually expanding it? I think the fact $\dfrac{1-x^3}{1-x}=1+x+x^2$ may help. But can't use it!
2
votes
1answer
43 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
1
vote
1answer
34 views

Upper bound for ${n \choose cn}$

Is it true that for any $0<c<1/2$ and sufficiently large $n'$, there exists a $d <2$ such that ${n \choose cn} < d^n$ for all $n>n'$? Clearly we have to assume $cn$ is an integer. I ...
5
votes
2answers
98 views

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is? I tried this: ...
0
votes
1answer
37 views

Generating functions of form $\sum_{n=0}^\infty a_n x^{kn}$

Let's consider generating function $$F(x) = (1+x)^r = \sum_{n=0} \binom{r}{n} x^n$$ And another generating function $$G(x) = (1+x^2)^r = \sum_{n=0} \binom{r}{n}x^{2n}$$ Please note those 2 functions ...
0
votes
1answer
41 views

Binomial coefficient - first two terms, proof of inequality

I've seen the following and I'm not sure whether it is true or not, and if yes, why it holds. $(1-p)^x \geq 1-p x$ for $p\in (0,1)$ and $x>0$. Do I need some additional Information to prove ...
1
vote
3answers
68 views

Calculate $\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}$

For $p \in [0,1]$ calculate $$S =\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}.$$ Since $$ (1-p)^{n-k} = \sum_{j=0}^{n-k} \binom{n-k}{j} (-p)^j, $$ then $$ S =\sum_{k=0}^n \sum_{j=0}^{n-k} k ...