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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divide 6 people in group of 2 in same size

Exercise: divide 6 people in group of 2 in same size. My solution: The exercise tells us to calculate the combination without repetition. If I start by calculating the number of ways to select how ...
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1answer
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proof: $\sum\limits_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$

Let $n ≥ 0$ and $k ≥ 0$ be integers. 1) How many bitstrings of length $n + 1$ have exactly $k + 1$ many $1$s? 2) Let $i$ be an integer with $k ≤ i ≤ n$. What is the number of bitstrings of length $n ...
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1answer
38 views

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$ Should I use the formula $C(n,k) = n!/[k!(n-k)!]$? And what is the solution of this problem?
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3answers
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Binomial coefficient problem

I still haven't quite realized how to solve binomial coefficient problems like this, can someone show me an elaborated way of solving this? I need to write this expression in a more simplified way: ...
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1answer
22 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
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3answers
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Proof by induction, binomial coefficient

I have to make the following proof: $${\sum\limits_{k=1}^n}{k}{n\choose k} = n2^{n-1}$$ Base case, $n = 1$: $${\sum\limits_{k=1}^{1}}{k}{1\choose k} = 1 = 1\cdot2^0=1$$ Inductive Hypothesis: for ...
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3answers
54 views

Proof of non divisibililty of $\binom{n}{r}$ with a prime $p$

I came across this : "It is possible to show that if $p$ is prime, $\binom{n}{r}$ is not divisible by $p$ if and only if the addition $r + (n-r)$, when written in base $p$, has no carries. This means ...
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Prove by induction on the binomial coefficient n choose m …

Prove by induction on $n$ that the binomial coefficient $\begin{pmatrix}n\\m\end{pmatrix}$ is the number of subsets of $I_{n}$ having size equal to $m$. The solution is as follows: So far it can be ...
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5answers
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how to prove: $\sum\limits_{k=1}^n k\binom{n}{k}=n \cdot 2^{n-1} $ [duplicate]

need help to prove this: $\sum\limits_{k=1}^n k\binom{n}{k}=n \cdot 2^{n-1} $ where $n$ is integer $\geq 1$. Question also said taking the derivative of $(1 + x)^n$ would be helpful which I've found ...
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0answers
59 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
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330 views

Proving the sum of squares of sine and cosine using the Cauchy product formula

Here are the power series of sine and cosine: $$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$$ and $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$$ How can it be ...
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3answers
141 views

The maximum of $\binom{n}{x+1}-\binom{n}{x}$

The following question comes from an American Olympiad problem. The reason why I am posting it here is that, although it seems really easy, it allows for some different and really interesting ...
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4answers
88 views

Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
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1answer
37 views

What is $\binom{n}{-k}$?

What is $\binom{n}{-k}$ ? If $n,k\ge0$ In Wikipedia there's a case where $n$ is negative and not $k$ But if Pascal's rule still holds, I get for example for $k=0$; ...
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2answers
27 views

Binomial Expansion where N is negative

Comparing the formula for regular binomial expansion (n>1): $(a+b)^n=a^n + \binom{n}1a^{n-1}b + \binom{n}2a^{n-2}b^2 +...$ to binomial expansion for negative indices, (n<1): $(1+x)^n= 1 + nx + ...
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1answer
36 views

Proof By Induction Using Binomial Coefficients

I'm having a really hard time with this proof by induction: Prove this formula by induction: $1^2 + 2^2 + 3^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6}$. Easy enough, right? Wrong. I have to do it using ...
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1answer
23 views

Finding the coefficient of a power series

How would I find the coefficient of: $[x^{10}]x^6(1-2x)^{-5}$ I know that I can simplify this as follows: $[x^4](1-2x)^{-5}$ and that generally the following formula would be used to solve this: ...
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0answers
41 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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1answer
23 views

Binomial coefficients bounded by entropy exponential

So I'm trying to prove that for $\frac{1}{2}< x \leq 1$ we have $$\sum_{\lceil nx \rceil}^{n}{n \choose k} \leq 2^{nh(x)}$$ I've managed to prove that $$\sum_{0}^{\lfloor nx \rfloor}{ n\choose ...
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4answers
48 views

Limit of quotient of summations involving special binomial coefficients

Find the limit, when $n$ tends to infinity, of $$ \frac{\displaystyle\sum_{k=0}^n\binom{2n}{2k}3^k} {\displaystyle\sum_{k=0}^{n-1}\binom{2n}{2k+1}3^k} $$ Please Help Me to solve the problem ...
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0answers
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Limit of $\frac{1}{n}\log({n\choose np})$ without using Stirling's formula

I am trying to evaluate the following limit: $$ \forall p\in(0,1) ,\lim_{n\rightarrow \infty} \frac{1}{n}\log{n\choose \lfloor np \rfloor} =H(p),$$ where $\lfloor x\rfloor$ means the greatest integer ...
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1answer
40 views

$\sum_{i=0}^{k} \binom{m}{i}\binom{n}{k-i} =\binom{m+n}{k}$ [duplicate]

I'm trying to show that the equality $$\sum_{i=0}^{k} \binom{m}{i}\binom{n}{k-i} =\binom{m+n}{k}$$ Is true. I know it is since there is a good combinatorical argument for it. If we have a group of ...
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44 views
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38 views

How to write binomial sum into integral [closed]

\begin{align} & \sum_{s=j}^n \sum_{r=i}^s \frac{n!}{r!(s-r)!(n-s)!}p_1^r(p_2-p_1)^{s-r}(1-p_2)^{n-2} \\[8pt] = {} & \int_0^{p_1} \int_{r_1}^{p_2} \frac{n!}{(i-1)!(j-i-1)!(n-j)!} \times ...
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0answers
34 views

Closed-form solution for $f(n) = \sum_{k>0}\binom{n}{2k}x^{k}$ without $\sqrt{x}$

Is it possible to reformulate the expression $$ (1+\sqrt{x})^n + (1-\sqrt{x})^n $$ in the form that contains no square roots of $x$ and no iterative sums (i.e. can be computed in constant time)? ...
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3answers
65 views

Showing ${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}$

Prove that for integers $n \geq 0$ and $a \geq 1$, $${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}.$$ I figured I'd post this question, ...
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3answers
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Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
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2answers
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On ${-1 \choose 0}=1$, can I assume that $\frac{(-1)!}{(-1)!}=1$?

I've had to evaluate ${-1 \choose0}$ and then I discovered the following: $${-1 \choose0}=\frac{(-1)!}{(-1)!0!}=\frac{(-1)!}{(-1)!}=1$$ Can I assume that $\frac{(-1)!}{(-1)!}=1$?
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1answer
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Binomial coefficients and cosin

In this question the user ask to prove the next identity: $$1 + 4\cos{2\theta} + 6 \cos{4\theta} +4\cos{6\theta}+ \cos {8\theta} =16\cos{4\theta} \cos^4 \theta$$ I realized the terms in the left ...
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1answer
31 views

What is the smallest value beside 1 of a binomial with two integer values > 0? [closed]

I'm searching for the smallest possible value of a binomial(a, b) where a >= b and both values are greater than ...
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1answer
28 views

sum of a binomial coefficient [duplicate]

Trying without success to solve the following: what is the sum of $\binom{80}{0}-\binom{80}{1}+\binom{80}{2}-\binom{80}{3}...-\binom{80}{79}+\binom{80}{80}$ any help will be greatly appreciated
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What is the coefficient of $x^{17}$ in the formula $(x^2+x)^{15} $?

What is the coefficient of $x^{17}$ in the formula $(x^2+x)^{15} $? Any idea how to solve this using the binomial coefficient formula?
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1answer
32 views

Calculating the sum of a binomial coefficient series

Calculate this: $$\bigl(\begin{smallmatrix} 80 \\0 \end {smallmatrix}\bigr)-\bigl(\begin{smallmatrix} 80 \\1 \end {smallmatrix}\bigr)+\bigl(\begin{smallmatrix} 80 \\2 \end ...
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3answers
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Evaluate: $\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$

Evaluate: $$\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$$ Attempt $S_2=\frac {n!}{(n-2)!}$ $S_3=\frac {n!}{(n-3)!}$ $S_4=\frac {n!}{2(n-4)!}$ $\vdots$ $S_{n-1}=\frac {n!}{1!(n-3)!}$ $S_n=\frac ...
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0answers
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Question on binomial coefficient power

Is there an expression to $$2^{\sum_{i=1}^{k}\binom{n}{i}}-2^{\sum_{i=1}^{k-1}\binom{n}{i}}?$$ Also is this expression solution to question in link Number of degree $k$ functions?
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0answers
35 views

Summation with Binomial Coefficients, $\sum (-1)^k \binom{m_1}{k} \binom{m_2}{k} $

I have trouble doing this summation: $$ \sum_{k=0}^{\min(m_1,m_2)} (-1)^k \binom{m_1}{k} \binom{m_2}{k} $$ where $m_1$ and $m_2$ are positive integers. Can someone help?
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Showing $\binom{2n}{n} = (-4)^n \binom{-1/2}{n}$

Is there a proof for the following identity that only uses the definition of the (generalized) binomial coefficient and basic transformations? Let $n$ be a non-negative integer. $$\binom{2n}{n} = ...
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1answer
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How to find the value of the following items summed up together?

How to find the value of the following items summed up together? ...
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230 views

When $\frac{C(n, k)}{n^{k-1}} > 1$?

I came across this while considering the subset sum problem in relation to another problem. Define the ratio, $$R(n,k) = \frac{C(n, k)}{n^{k-1}} = \frac{\binom n k}{n^{k-1}}$$ and the integer ...
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Question about Binomial Sums [duplicate]

Prove that for any $a \in \mathbb{R}$ $$\sum_{k=0}^n (-1)^{k}\binom{n}{k}(a-k)^{n}=n!$$ I rewrote the sum as $$\sum_{k=0}^n \left((-1)^{k}\binom{n}{k} \sum_{i=0}^n (-1)^{i}a^{n-i} k^{i} ...
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3answers
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Find the sum $\sum_{j=0}^{n}\binom{4n}{4j}$

Find the sum of the series $$\binom{4n}{0}+\binom{4n}{4}+\binom{4n}{8}+\ldots+\binom{4n}{n}=\sum_{j=0}^{n}\binom{4n}{4j}.$$ My approach is to consider $(1+x)^{4n} = \sum_{j=0}^{4n}\binom{4n}{j}x^j.$ ...
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0answers
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Integers in the form $\sum_{j=0}^n a_j2^j3^{n-j}$

Let $n>0$ be an integer. Let also $a_j$ be some integer in the set $\{0,1,\ldots,\binom{n}{j}\}$ for all $j=0,1,\ldots,n$. Then, how many integers can be written in the form $$2^n ...
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1answer
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Hillman and Hoggat's Binomial Generalization

In proving Gould's "Star of David" conjecture, Hillman and Hoggat generalized the binomial coefficient. First, they demand that $a_n$ be a sequence with the two properties that $$\gcd(a_m, a_n) \mid ...
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2answers
86 views

Combinatorial proof for $ \sum _{r=1} ^n r^3 \binom nr = n^2(n+3) 2^{n-3}$

Find the combinatorial proof for $$ \sum _{r=1} ^n r^3 \binom nr = n^2(n+3) 2^{n-3}$$ After proving it using algebra, I'm unable to find a combinatorial argument for the above statement. Help ...
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59 views

How many routes are there that pass through at most one congested intersection

I am trying to solve the following problem, but i am not quite sure how to attack. Problem Description A taxi drives from the intersection labeled A to the intersection labeled B in the grid of ...
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3answers
79 views

Finite summation with binomial coefficients, $\sum (-1)^k\binom{r}{k} \binom{k/2}{q}$

I came across the following finite sum involving (generalized) binomial coefficients: $$ 2^q \sum_{k=0}^r \binom{r}{k} \binom{k/2}{q} (-1)^k .$$ Putting this into Mathematica gives me: $$ (-1)^q ...
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votes
5answers
135 views

How to find $ \binom {1}{k} + \binom {2}{k} + \binom{3}{k} + … + \binom{n}{k} $

Find $$ \binom {1}{k} + \binom{2}{k} + \binom{3}{k} + ... + \binom {n}{k} $$ if $0 \le k \le n$ Any method for solving this problem? I've not achieved anything so far. Thanks in advance!
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1answer
23 views

$X_n = \sqrt[k]{n^{p}+an^{q}+1}-\sqrt[k]{n^{p}+bn^{q}+1} $

For what given p and q below sequence is bounded? $X_n = \sqrt[k]{n^{p}+an^{q}+1}-\sqrt[k]{n^{p}+bn^{q}+1} $ where $0\leq q<p$ and $a\ne b$ My try ...
2
votes
4answers
37 views

Find the value of $ \sum _{r=0} ^{2n} r ( ^{2n}C _r) ( \frac 1{r+2} ) $

Find the value of $$ \sum _{r=0} ^{2n} r ( ^{2n}C _r ) ( \frac 1{r+2} )$$ In order to solve this I am trying to make the term(s) of the series independent of $r$. However I'm unable to solve ...
3
votes
3answers
75 views

A combinatorial identity: $\sum _{i + j = k} (-1)^i {n \choose i} {n + j - 1 \choose n - 1 } = 0 $

I proved this combinatorial identity while doing some linear algebra. For any positive integer $k$, $$ \sum _{i + j = k} (-1)^i {n \choose i} {n + j - 1 \choose n - 1 } = 0 $$ I was wondering what ...