Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

learn more… | top users | synonyms (1)

1
vote
3answers
41 views

Let $a$ and $b$ be the coefficient of $x^3$ in $(1+x+2x^2+3x^3)^4$ and $(1+x+2x^2+3x^3+4x^4)^4$ respectively.

Let $a$ and $b$ be the coefficient of $x^3$ in $(1+x+2x^2+3x^3)^4$ and $(1+x+2x^2+3x^3+4x^4)^4$ respectively.Find $(a-b).$ I tried to factorize $(1+x+2x^2+3x^3)$ and $(1+x+2x^2+3x^3+4x^4)$ into ...
-1
votes
2answers
51 views

How to prove that $(n+1)\binom{n}{k}=(k+1)\binom{n+1}{k+1}$? [closed]

How to prove that for the integers $k,n$ where $k \leq n$ the following holds: $$(n+1)\binom{n}{k}=(k+1)\binom{n+1}{k+1}$$
4
votes
2answers
40 views

Let $f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}$.If $f(n)=2047,$ then find the value of $n.$

Let $f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}$.If $f(n)=2047,$ then find the value of $n.$ $f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}=\sum_{k=0}^{n}\binom{k}{0}+\sum_{k=1}^{n}\binom{k}{1}+\...
-2
votes
3answers
71 views

Prove algebraically that ${n\choose k}=\frac{n(n-1)…(n-k+1)}{1\cdot 2\cdot …\cdot k}$

From the definition of binomial coefficient, $${n\choose k}=\frac{n!}{k!(n-k)!}\Rightarrow \frac{n!}{k!(n-k)!}=\frac{n(n-1)...(n-k+1)}{k!}$$ $$\Rightarrow \frac{n!}{(n-k)!}=n(n-1)...(n-k+1)$$ Could ...
1
vote
0answers
25 views

On a theorem of Hensel

In the paper Binomial coefficients modulo prime powers, Andrew Granville state the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
1
vote
0answers
25 views

Multinomial coefficients modulo a prime

Let $p$ be a prime and let $m \geq 1$. Lucas' theorem implies that the binomial coefficient ${p^m-1 \choose k}$ is not divisible by $p$ for any $0 \leq k \leq p^m-1$. I wonder if something similar ...
3
votes
2answers
31 views

Prove $\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2}=\binom{n+1}{4}$

How to prove $\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2}=\binom{n+1}{4}$? I tried writing it as an AGP as following: $$\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2} = \textrm{coefficient of } x^...
1
vote
2answers
42 views

Evaluate $\binom{m}{i} - \binom{m}{1}\binom {m-1}{ i} + \binom{m}{2}\binom{m - 2}{i} - \ldots + (-1)^{m-i} \binom{m}{m-i}\binom{ i }{i} $

Evaluate the expression $$\binom{m}{i} - \binom{m}{1}\binom {m-1}{ i} + \binom{m}{2}\binom{m - 2}{i} - \ldots + (-1)^{m-i} \binom{m}{m-i}\binom{ i }{i} $$ I'm really stumped about trying to get ...
1
vote
1answer
39 views

How can I calculate $\sum_{k=1}^{n-1}\binom{n-1}{n-k}$?

I would like to know if I can calculate a closed expression for $$\sum_{k=1}^{n-1}\binom{n-1}{n-k}$$ This sum is equals to: $$1+(n-1)+(n-1)(n-2)+(n-1)(n-2)(n-3)+\ldots+(n-1)(n-2)/2+(n-1)$$
0
votes
0answers
28 views

Proof of the diagonalization of the probability matrix of the sum of two binomial distributions

I am analysing a statiscal problem where a vector $X\in\mathbb Z_{\geq 0}^{n+1}$ is probabilistically transformed according to $x_i \mapsto \mathrm{Binomial}(x_i, 1/2+\epsilon/2) + \mathrm{Binomial}(n-...
1
vote
1answer
18 views

Is my intuition about this statistics problem sensible?

I'm trying to improve my knowledge of statistics and develop my intuition for solving statistical problems. While doing so I've worked on the following exercise: There are 20 players in a checkers ...
1
vote
2answers
97 views

Verify the following identity algebraically

Verify the following identity algebraically (writing out the binomial coefficients as factorials).$${n \choose k}{k \choose m} = {n \choose m}{n-m \choose k-m}$$ So far, these are my steps: $$\frac{...
5
votes
1answer
76 views

Reference request for an identity involving binomial coefficients

The identity is $$\sum_{i\ge 0}(-1)^i \binom{\frac{s^k + s^{-k} - 10}{4}}{i}\binom{\frac{s^k + s^{-k} - 10}{4}+4}{\frac{gs^k + 2g^{-1}s^{-k} - 4}{8}-i}= 0$$ where $k\gt 1$ , $s = 3+2\sqrt{2}$ ...
0
votes
0answers
52 views

Sum of Two Double Sums

Suppose $ r^{2}=4ab $. Show that over the complex number field: $$ \left( \sum_{k=0}^{l}{\sum_{m=0}^{l-k}{\binom{l}{k}\binom{l-k}{m}a^{l-k-m}b^{m}(-1)^{m}r^{k-1}i^{k-1} \left( \frac{2a-(-1)^{k}\left(...
0
votes
1answer
47 views

Find the sum of the series .

The general term of the series is $\sum_{r=0}^{100}\binom{500}{r}\binom{500-r}{400}2^{100-r}$ What I tried to think that this series was an expansion for a series inside a series but the thing that ...
2
votes
0answers
38 views

Number of ways of selecting atleast one book from 9 different books of 10 copies each.

Number of ways of selecting atleast one book from 9 different books of 10 copies each. Let $x_i$ denote the number of copies selected from $i^{\text{th}}$ type of book. $$\sum_{i=1}^9 x_i\le 90$$ I ...
1
vote
2answers
55 views

Variation on Vandermonde's identity

How can you show that $$ \binom{2n}{n}^2 = \sum_{m=0}^{n} \binom{2n}{2m} \binom{2m}m \binom{2n-2m}{n-m} $$? I was fooling around with random walks, and apparently both expressions are supposed to be ...
1
vote
3answers
78 views

Criticize my math when I attempt to find the coefficient of $x^2y^6$ in the expansion of $(x+2y^2)^5$

So I look around this site and my textbook (Richmond&Richmond, discrete math) and I know I'm in the right direction but I'm also sure I am doing it wrong. Original Question: find the coefficients ...
2
votes
0answers
71 views

Spread Polynomials Identity (Rational Trigonometry)

Show that if $ n=2l+1 $ is an odd natural number then $$ S_{n}\left( s \right)=s\left( \binom{n}{1}\left( 1-s \right)^{l}-\binom{n}{3}\left( 1-s \right)^{l-1}s+\binom{n}{5}\left( 1-s \right)^{l-2}s^{2}...
1
vote
1answer
38 views

How to differentiate $\big[\sum_{k=0}^n\binom{n}{k}a^{n-k}b^k\big]$ with respect to $n$ without summing?

I know the answer because of the following derivation: $$ {d\over d n}\left[\sum_{k=0}^n\binom{n}{k}a^{n-k}b^k\right]= {d\over d n}(a+b)^n=(a+b)^n\log{(a+b)} $$ The way of calculating it is to ...
1
vote
1answer
28 views

Is there a way to mathematically express the sum of all combinations of a set of constants?

Suppose that I have a set of $n$ constants. I want to find the sums of all product combinations of length $i = n \rightarrow 0$ Using $n=5$ as an example: $C = {a,b,c,d,e} $ $C_5= abcde$ $C_4= ...
3
votes
0answers
63 views

A combinatorial proof for a bound on diagonal Ramsey numbers

I wish to prove $R(p,p)\leq\frac{2^{2p-2}}{\sqrt{p}}$ combinatorially. I have proved this algebraically through the definition of the binomial coefficient but I would much prefer a proof from ...
3
votes
5answers
48 views

Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal

Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal. My attempt: $\displaystyle \binom{...
4
votes
2answers
43 views

Trouble understanding how this identity is derived: $\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$

$$\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$$ The $-a-1$ is throwing me off. Can anyone help me understand this identity. I have tried letting $m=-a-1$ and then applying the binomial theorem,...
1
vote
1answer
100 views

Encyclopedia of Integer Sequences - Formula

I am trying to reproduce the following sequence (https://oeis.org/A062734): ...
0
votes
0answers
16 views

rows of pascals triangle as powers of 11 in different numeral systems

It's not too difficult to see that (and understand why) in a base n system the ciphers of $(11_n)^k$ are equivalent to the k-th (0-indexed) row of pascals triangle until one of the numbers becomes ...
1
vote
1answer
53 views

Formula for a geometric series weighted by binomial coefficients (sum over the upper index):$\sum_{i=0}^L {n+i\choose n}\ x^i =\ ?$

The binomial sum is $$\sum\limits_{i=0}^n {n\choose i}\ x^i = (1+x)^n,$$ where $\displaystyle{n\choose i}=\frac{n!}{(n-i)!i!}.$ Is there a corresponding formula when you sum over the upper index of ...
1
vote
2answers
32 views

Probability that binomial random variable is greater than another

Let $X$ and $Y$ be two independent random variables with respective distributions $B(n+1,\frac{1}{2})$ and $B(n,\frac{1}{2})$. I am trying to determine $\mathbb{P}(X>Y)$. So far, I have written ...
1
vote
2answers
64 views

Prove ${n \choose k} = {n \choose k-1}\frac{n-k+1}{k}$

I am looking to prove, by induction, the following equality:$${n \choose k} = {n \choose k-1}\frac{n-k+1}{k}$$ From Pascal's identity, I know we have that $${n+1 \choose k} = {n \choose k} + {n \...
1
vote
1answer
31 views

Complex counting problem

A code has five symbols and each symbol is composed of a letter, a number, and a color. (Ex.{a,3,black} is a symbol) there are $p$ letters, $q$ numbers and $r$ colors to choose from. a code ...
0
votes
1answer
29 views

Binomial Coefficient Explanaition

Let $n\in\mathbb{N}$ and let $k\in\{0,\ldots,n\}$. Explain why it follows from $$\binom nk=\frac{n}{1}\times\frac{n-1}{2}\times\frac{n-2}{3}\times\cdots\times\frac{n-k+1}{k}$$ that $$\binom nk=\frac{...
0
votes
1answer
21 views

Binomial Coefficient Pattern [closed]

Let $n$ $\epsilon$ N and let $k$ $\epsilon$ {0,...,n}. Explain why it follows from ${n \choose k}$ = ${n \choose k-1}$$\frac{n-k+1}{k}$ that ${n \choose k}$ = ($\frac{n}{1}$)($\frac{n-1}{2}$)($\frac{n-...
0
votes
0answers
31 views

Evaluating a sum on binomial coefficients

I'm reading Casella's and Berger's Statistical Inference. On page 239 they gives a claim that $$\sum_{x=0}^{330}\binom{300+x-1}x\left(\frac{1}{2}\right )^{300}\left (\frac{1}{2}\right )^x\approx 0....
2
votes
2answers
33 views

Calculation of the limit of the difference of binomial coefficients

This question pertains to harmonic analysis on spheres. Let $H_d$ = {homogeneous, total degree $d$ harmonic polynomials in $\mathbb{C}[x_1,\dots,x_n]$} Given that the Dimension of $H_d = \...
2
votes
0answers
22 views

Evaluate the sum $\sum_{i=0}^n \binom{n}{i}^2 i^k$

Given a positive integer $k$, is it possible to evaluate the following sum? $$ \sum_{i=0}^n \binom{n}{i}^2 i^k\,\,\,? $$ [I know just for $k=0$ the sum is $\binom{2n}{n} \approx 4^n/\sqrt{n}$..]
2
votes
1answer
14 views

Factorials and equivalency

I am not sure if this would be a proper title because I am a bit confused, but I was reading about proving Pascal's Triangle, and there was a proof on here I was following everything that was ...
-1
votes
3answers
30 views

Binomial coefficients algebra

I am trying to prove $$\binom{n+1}{k} = \binom{n+1}{k-1}\frac{n-k+2}{k}$$ by using the following four equations: \begin{align*} \binom{n + 1}{k} & = \binom{n}{k} + \binom{n}{k - 1}\\ \binom{n + ...
2
votes
0answers
34 views

Pascal's triangle induction proof

I am trying to prove $$\binom{n}{k} = \binom{n}{k-1}\frac{n-k+1}{k}$$ for each $k \in \{1,...,n\}$ by induction. My professor gave us a hint for the inductive step to use the following four equations:...
3
votes
2answers
35 views

Coefficient of $x^{n-1}$ in the given expansion

The problem I am facing is that with each term, number of ways to achieve $x^{n-1}$ is increasing, so it is getting very difficult to club all the cases together. Please provide some insight.
0
votes
1answer
70 views

Prove the following equality: $ \sum_{i=0} ^n j {n \choose j} = n 2^{n-1} $ [duplicate]

I'd like some help. My first idea was to use induction, but then I get stuck. The base case works just fine, as you'd imagine, and then... If $ \sum_{i=0} ^n j {n \choose j} = n 2^{n-1} \rightarrow ...
1
vote
1answer
26 views

On a corollary of Kummer theorem

I faced this problem when I learn Kummer theorem : Let $p$ be a prime number and $m, n$ are two positive integer such that $p\nmid m$ and $p^k\mid n$ for some positive integer $k$. Prove that $p^k\...
2
votes
0answers
19 views

On a corrolary of Lucas theorem

In this paper, the author stated that we can use Lucas theorem to prove that $$\binom{ap^f+r}{m}\equiv \binom{r}{m}\ (\text{mod}\ p)$$ where $0\le r, m < p^f$, $a\ge 0$. I don't know how to use ...
4
votes
1answer
47 views

Find the remainder when divided by $2017$

Given $2017$ is a prime number. Let $S=\sum_{k=0}^{k=62} \binom{2014}{k}$. Find the remainder when $S$ is divided by $2017$. I am unable to simplify the expression for $S$. Need some hints. Thanks.
0
votes
0answers
49 views

Why is $\binom{n}{3}p^3\left(1+\frac{3(np)^2}{n}\right) \le \frac{10(np)^5}{n}$

Let $Z = \binom{n}{3}p^3$ and $w=np$. Here $p$ represents a probability. Why is $Z\left(1+ \frac{3w^2}{n}\right) \le \frac{10w^5}{n}$? Edit: There are some other condtions that I didn't think were ...
10
votes
3answers
191 views

Is there a closed form or approximation to $\sum_{i=0}^n\binom{\binom{n}{i}}{i}$

I tried to calculate the sum $$ \sum_{i=0}^n\binom{\binom{n}{i}}{i} $$ but it seems that all my known methods are poor for this. Not to mention the intimate recursion, that is $$ \sum_{i=0}^n\binom{\...
2
votes
1answer
52 views

Divisibility of Binomial Coefficients by a Composite Number [duplicate]

I am aware of proof of divisibility of binomial coefficients of a prime $p$. I've seen it is easy to show that when $0<k<p$ $$\binom{p}{k}\equiv 0 \mod p$$ Can there be anything stronger. ...
1
vote
2answers
53 views

How do you determine the following coefficient?

$$[x^n]\frac{(1+x)^n}{(1-x)}$$ It seems pretty simple but I can't seem to find it. I tried rewriting it as the product of two sums. $(1+x)^n=\sum_{k=0}^n {n \choose k}x^n$ and $\frac{1}{1-x}=1+x+x^2+....
0
votes
2answers
55 views

What is the sum of binomial-coefficients multiplied by i?

$$\sum_{i=0}^n {n \choose i}i$$ I know this is equivalent to $\sum_{i=0}^n \frac{n!i}{i!(n-i)!}$, but the factorial prevents me from solving this easily.
2
votes
1answer
49 views

Algebraic identity involving powers of twin primes

Yesterday, I verified that, if $a$,$b$ and $c$ are real numbers such that $a+b+c=0$, then $$\frac{a^5+b^5+c^5}{5}=\frac{a^3+b^3+c^3}{3}\cdot\frac{a^2+b^2+c^2}{2}$$ and $$\frac{a^7+b^7+c^7}{7}=\frac{...
2
votes
2answers
187 views

Subtracting even and odd binomial coefficients?

What number do I get if I subtract the binomial coefficients $n\choose k$ with an even $k$ from those with an odd $k$, where $n$ is fixed? Am I supposed to subtract a binomial coefficient with an even ...