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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Using binomal theorem, show that $99^5+1$ is divisible by $100$ [on hold]

This is my homework. Please state working clearly and explain how you get it Thank you.
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1answer
38 views

Parallelogram inequality in Pascal's triangle

While looking at Pascal's triangle I noticed $$\binom{n}{k} > \binom{n+1}{k-1} \tag{*}$$ for "most" pairs $(n, k)$ on the left half i.e. for $1 \le k \le n/2$. Question. Define $\kappa(n) = ...
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i want solution of this problem? [on hold]

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1answer
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calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$

Is there any formula for calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$?
1
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1answer
25 views

Is this true that $\sum_{v=0}^k (-1)^v{a+v-1 \choose v}{b+k-v-1 \choose k-v}$ is the coefficient of $t^k$ in $(\frac{1}{1+t})^a(\frac{1}{1-t})^b$

I was reading a paper, in which the author assumed that $$\sum_{v=0}^k (-1)^v{a+v-1 \choose v}{b+k-v-1 \choose k-v}$$ is the coefficient of $$t^k $$ in ...
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0answers
44 views

how find an event with probability $6.6p^2q$ [closed]

How to find events with the following probabilities $6.6p^2q$,$6.7p^2q$,$6.5p^2q$,$3.4pq$ using repeated bernoulli trials with success probability p[$\backepsilon$ $0<p<1$]?
3
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2answers
191 views

Find the probability of getting one diamond and one spade in a five-card hand, using binomial coefficients.

A five card hand is dealt at random from a standard $52$ card deck. Let $X = \text{# spades}$ and $Y = \text{# diamonds}$. Find $P(X = 1\text{ and }Y =1)$. Leave your answer as a ratio of ...
7
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1answer
352 views

Inequality involving factorial $\binom nk<(en/k)^k$

I am trying to prove following inequality: $$\binom{n}{k}<(en/k)^k$$ I tried Stirling approximation but I could not get anything further. Then I get $$\binom{n}{k}\approx \frac{\sqrt{2\pi ...
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0answers
22 views

Sum of first $\frac {n}{2}$ binomial coefficients [on hold]

What is the sum of first $\frac {n}{2}$ binomial coefficients, if $n$ is an even number? Also what is the sum of first $\frac {n-1}{2}$ binomial coefficients if $n$ is odd?
1
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1answer
36 views

Vandermonde's Convolution special case.

I am not able to show this case of Vandermonde's Convolution without using induction. Can someone help me? $$ \binom{n}{m} = \sum_{k=0}^{m} \binom{n-p}{m-k} \binom{p}{k}. $$ I thank now.
4
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4answers
285 views

Sum of sum of binomial coefficients

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
4
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1answer
69 views

The sequence $(-1)^n\binom{\alpha-1}{n}$ converges.

I need to show that for $n \in \mathbb N_0$ and $\alpha \ge 0$ the sequence $(-1)^n\binom{\alpha-1}{n}$ converges. It can be shown that the sequence convereges to zero using a theorem claiming that ...
1
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1answer
26 views

Binomial expansion extension to negative powers

I know that: $$\sum_{k=0}^n {n \choose k}a^{n-k}b^k = (a+b)^n$$ But how is this extended to negative powers, for example, I came across the following line of maths, which I struggle to understand: ...
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How to calculate the total number of dissimilar terms (terms having different powers in x)… [closed]

The question is that: Calculate the total number of dissimilar terms i.e different powers of x in the binomial expansion of $(1-x+x^{-2}-x^2)^{50}$. I have no idea how to evaluate such type ...
0
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Multivariable recurrence: Solving $c(n,k) = c(n-1,k) + c(n-1,k-1) = \binom{n}{k}$ by algebraic methods.

Let $(c_{n,k})_{n,k=0}^{\infty}$ be defined by $c(0,0)=1$, $\:\:c(0,k)=0 \:\: \forall \: k > 0$ $$c(n,k) = c(n-1,k) + c(n-1,k-1) \:\:\: \forall \: n \geq 1$$ I can show that the ...
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1answer
63 views

Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
1
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2answers
35 views

binomial coefficient where k > n

For solving binomial coefficients we have use from formula $\frac{n!}{k!(n-k)!}$ This formula only works if n > k. What happens if n < k? Is there another formula we need to use?
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6answers
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You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life.

This is from the lecture notes in this course of discrete mathematics I am following. The professor is writing about how fast binomial coefficients grow. "So, suppose you had 2 minutes to save your ...
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0answers
23 views

arranging $n$ objects of one kind and $m$ objects of other kind in a row

Why are there precisely $\binom{m+n}{n}$ ways of arranging $M$ objects of one kind and $N$ objects of other kind in a row?
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0answers
30 views

Logic - Binomial Theorem

I could use some assistance with understanding this problem. I understand that there are ${n}\choose{k}$ is a representation of ${n}\choose{k}$ ways to choose k elements from a set of n elements, ...
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2answers
44 views

Alternating series of compositions of triangular numbers

I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls. Each time I add a ball to $S$, it may dislodge another ball with probability proportional to ...
4
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Find the sum of $\binom{2007}{0}+\binom{2007}{4}+…+\binom{2007}{2004}$ [duplicate]

Find the sum of $$S=\binom{2007}{0}+\binom{2007}{4}+\binom{2007}{8}+...+\binom{2007}{2004}$$ My work so far: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ ...
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1answer
21 views

Closed form for binomial sum with absolute value

Do you know whether the following expression has a (nice) closed form or a close enough approximation? $$\frac{1}{2^n}\sum_{k=0}^{n} \binom{n}{k}|n-2k|$$ Thanks a lot :) Cheers, M.
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22 views

On a certain series of cosines

For natural numbers $p$ and $q$, compute the value of $$\displaystyle \sum_{k=0}^{q-1} \cos^{p} \left(\dfrac{2\pi k}{q}\right).$$ I got the answer $$\dfrac{q}{2^p} \sum_{l=1}^{p} \binom{p}{l} ...
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Intuitive explanation of Extended binomial coefficient

We all are familiar with the following formula - $$\dbinom{n}r = \dfrac{n!}{(n-r)! \space r!} \space\space \space ; \space \space n>r$$ This is the binomial formula where $n$ and $r$ are ...
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0answers
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Value of the binomial formula $\sum_{k=0}^n \binom{n}{k}k^n$?

Does anyone know the value of: $\sum_{k=0}^n \binom{n}{k}k^n$. I can't figure out how to derive out this formula from standard formulas
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Proving that ${p \choose r}$ is an integer for a prime $p$ and $0 < r < p, r \in \mathbb{Z}$ [duplicate]

I need to prove that given integers $p$ and $r$ such that $p$ is prime and $0 < r < p$, ${p \choose r} = \frac{p!}{r!(p-r)!} \in \mathbb{Z}$ As of now, I don't have any ideas on how to proceed. ...
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2answers
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Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...
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3answers
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Stirling on ${2n-1 \choose n}$

I'm trying to find an expression for $${2n-1 \choose n}$$ using Stirling's approximation $$k!\sim \sqrt{2\pi k}(\frac{k}{e})^k.$$ I see $${2n-1 \choose n}\approx ...
4
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2answers
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Prove that this system of linear equations generates $\left| \left( \begin{matrix} 1/2 \\ n \end{matrix} \right) \right|$ as a solution?

This infinite system of linear equations: $$ \begin{array}( 2x_1=1 \\ 3x_1+4x_2=2 \\ 4x_1+5x_2+6x_3=3 \\ \cdots \end{array} $$ In other words, this is particular case of a system: $$ \begin{array}( ...
2
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2answers
50 views

Catalan numbers formula derivation

I'm trying to follow a proof of the Catalan numbers being equal to $\frac{1}{n+1} {2n \choose n}$ from the recurrence relation $C_n = C_0C_{n-1}+C_1C_{n-2}+...+C_{n-2}C_{1}+C_{n-1}C_0$ Now it's seen ...
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1answer
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Generating function for multiset formula

It's said that the generating function for $g(x) = \sum_{d=0}^\infty {d+m-1 \choose m-1} x^d$ is equal to $\frac{1}{(1-x)^m}$. In the proof that I have seen it states that: By the geometric series, ...
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Smallest Parameter to Satisfy Exponentially Scaled Binomial Coefficient Inequality

Let $t$ be given, I am mainly interested in large $t$. Define $m(t)$ as below $$ m(t)=\min\left\{m: \sum_{k=0}^m \binom{t+k-2}{k} 2^{t+k} \geq 2^{2t}\right\}. $$ Is there a nice estimate for $m(t)$? ...
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The number $\binom{8}{4}$ is equal to the number of subsets of size 4 of the set $\{1, \dots, 8\}$

I was asked to proof if is true and give a counter example if it is false. However I prefer True. since all the numbers 1-8 insides the brackets are in the sets. I'm I correct?
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3answers
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How to sum binomial coefficients which are multiples of 3? [duplicate]

Basically $\sum_{i=0}^{33}\binom{99}{3i} $ I have read about this thread. I am looking for a conventional approach. Like the one we use in $\sum_{i=0}^{n}\binom{2n}{2i} = 2^{2n-1}$
2
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1answer
60 views

Summation involving binomial coefficients

I came across the following sum online and have spent awhile trying to compute it: $$\sum_{i=0}^{100} \binom{300}{3i}$$ Based on a pattern I noticed, the answer should be $\frac{2^{300}}{3}$ rounded ...
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3answers
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Find the middle number in the $29$th row in the Pascal's Triangle

This question is taken from the Singapore Mathematical Olmpiad training notes for Primary school. Find the middle number in the $29$th row of the Pascal's triangle. For example, the middle number ...
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1answer
555 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. ...
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Simpler form of binomial coefficients product

I am trying to find a simpler relation or an approximation of a product of binomial coefficients. This product is given by: $\Pi_{i=a}^{N-1}\binom{N}{i+1}$ Or if there is a starting point towards ...
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1answer
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binomial calculation method

I want solve this probability: For $p= 0.4$ $q=0.8$ $n= 20$ $1-P(5<x<11)$ = $1-\sum_{k=6}^{10} \binom{20}{k}(0.4)^k(0.6)^{20-k}. Wolfram Alpha -> = 0,2531$ Is calculation method ...
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Calculating summations concerning binominal coefficients

Since $$\sum_{j=0}^{m}\binom{k}{j}\binom{k}{m-j}=\binom{2k}{m},$$ what is the result of $$\sum_{j=0}^{m}\binom{k}{j}\binom{k}{m-j}(\frac{1}{3})^{j}=?$$ Here $i, j, k, m$ are integers.
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1answer
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Can't Remember a Book about Binomial Sums and Hypergeometry

Some time ago I had come across a website which had the online version of a book about techniques dealing with the solution of sums involving binomial coefficients, and something with the word ...
2
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1answer
37 views

Number of Elements in a Conjugacy Class of $S_N$ (Derivation)

Consider the conjugacy classes of the symmetric group $S_N$. Each conjugacy class consists of permutations that have the same cycle structure. We see that the number of possible cycle structures is ...
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1answer
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Choosing spread out elements

Is there an explicit formula that I'm missing, for the total number of choices from a set $S = \lbrace 1, 2 , \dots , n \rbrace$ such that for every choice $C \subset S$ the following holds: $\forall ...
2
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1answer
44 views

How to prove that $\binom{X+L-1}{L-1} \ge (X-L\times N)^{L-1}$?

I would like to prove the following expression: $$\binom{X+L-1}{L-1} \ge (X-L\times N)^{L-1}$$ , where $X$, $L$ and $N$ are positive integers. Please help me to prove with the following case. $X\ge ...
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Proving $\left(\binom{n}{k}\right)=\left(\binom{n-1}{0}\right)+\left(\binom{n-1}{1}\right)+\cdots+\left(\binom{n-1}{k}\right)$

Here, $\left(\binom{n}{k}\right)$ denotes the number of multisets in $N$ with length $k$. I can prove it using the fact that $\left(\binom{n}{k}\right) = \binom{n+k-1}{k}$ but I want another access. ...
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2answers
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How to prove $p^2 \mid \binom {2p} {p }-2$ for prime $p$?

How to prove $p^2 \mid \binom {2p} {p } -2$ for prime $p$? I have a hint: for $1 \le i \le p-1$, $p \mid \binom p i$. I cannot even start the proof. Please help.
4
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1answer
114 views

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$ I have no clue how to start. Any kind of help will be appreciated.
4
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2answers
468 views

This question involves Pascal's Triangle & Binomial Theorem. Full Question

Could anyone please help me on the following problem: Factorize the expression $P(n)=n^x-n$ for $x=2,3,4,5$ Determine if the expression is always divisible by the corresponding $x$. If divisible use ...
0
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2answers
42 views

Combinatorial identity involving binomial coefficients.

In order to conclude a proof (see last equality in B. Poonen's article), I need to establish the following identity: $$\forall (\ell,n)\in\mathbb{N}^2,\ell\leqslant n,\sum_{m=\ell}^n{n\choose ...