# Tagged Questions

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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### Double counting proof of binomial problem

The assignment is to prove the following assertion using the method of double counting and explaining which pairs were counted. $$\dbinom{n+1}{k+1} = \sum_{i = k}^{n} \dbinom{i}{k}$$ Left side is ...
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### Finding the coefficient of expansion

Question: Find the coefficient of $x^{11}$ in the expansion of:$$(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$$ The traditional way of doing this, as far as I know, is to first find the coefficient of each term ...
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### Of Balls in Bins in Different Sections with Caps

Problem: There are $19$ bins: $7, 5, 7$ in the left, centre and right sections respectively. There are $8$ balls, some or all of which are to be put into these bins with the following ...
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### Combinatorial identity / expected distance of random walk

I am struggling to verify the following identity. $$\binom{2m}{m} \frac{m}{2} = \sum_{j=1}^m j \binom{2m}{m+j}$$ I've tried induction, but I run into issues inside the sum. I can't see a ...
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### Error of Stirling’s approximation for Binomial with central limit theorem

So the question asks: Let $X_n$~Bin(2n,1/2),use Stirling’s approximation for $n!$ to show $P [X_n = n]$~ $1/√(πn)$ as $n→ ∞$, and show the error in the estimate for $P [X_n ≤ n]$, given by the central ...
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### Computing the coefficient of $x^n$ in the following expansion

The coefficient of $x^{-n}$ in the expansion of $\frac{2-3x}{1-3x+2x^2}$ is $a.)$ $(-3)^n - (2)^{\frac{1}{2}n -1}$ $b.)$ $2^n + 1$ $c.)$ $3(2)^{\frac{1}{2}n - 1} - 2(3)^n$ $d.)$ None of the ...
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### Closed form of a finite sum with binomial coefficients

In general, has $\sum_{k=a}^{b} \binom{n}{k}$ a closed form (with $0\le a\le b\le n$)?
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### Check whether or not a triangular number is triangular is the square-sum of two other consecutive triangular numbers

I'm trying to write a program that would tell me whether or not a triangular number, a number of the form $\frac{(n)(n+1)}{2}$ is the sum of the squares of two other consecutive triangular numbers. It ...
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### Coefficient of Multinomial kind of expression

How do I find the Multinomial coefficient of expression. For example $(x+y+z+w+6)^8$ let say I want the coefficient of xyzw. I know the answer in the simple case of $(x+3)^5$ , for $x^2$ it will ...
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### Is there a name for this combinatorial identity?

I found this identity in a textbook that I own but they did not name the identity and I had some trouble finding it online. Does anyone know the name of the identity and if I can find a resource about ...
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### Proving a binomial coefficient identity [duplicate]

I'm having some trouble with the following proof: $$\sum^k_{a=0} {{n}\choose{a}}{{m}\choose{k-a}} = {{n+m}\choose{k}}$$ I'm trying to prove this to learn a couple of things about the Pascal's ...
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### How in this world can I simplify this $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????

I have a problem, obviously. I am doing some maths and now I have to simplify this: $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????. But I just don´t know how ???? I´ve started simplifying by ...
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### Last digit of a number

I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...
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### Binomial Expansion - Finding the term independent of n.

The coefficient of $x^2$ in the expansion of $\left(1 + \frac x5\right)^n$, where $n$ is a positive integer, is $\frac 35$ . $(i)$ Find the value of $n$. $(ii)$ Using this value of $n$, find ...
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### Why is $\sum\limits_{k:|2k/l-m/l|\geq\epsilon/2}\frac{\binom{m}{k}\binom{2l-m}{l-k}}{\binom{2l}{l}}\geq 2e^{-\epsilon^2l/8}$

This is stated in an article on the uniform convergence of probabilities of events to their relative frequencies. The idea behind the question is that I have a measurement on a sample of size $2l$ ...
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### Binomial coefficient definition

Why is the definition of the binomial coefficient $${{m}\choose {r}}=\frac{m(m-1)\cdots(m-r+1)}{r!}$$ I'm not sure where the last term in the numerator came about. Why should there be a $+1$? ...
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### Problem with binomial coefficients?

I am trying to find the sum of $$\sum_{x=0}^{n-2}\left (\frac{1}{x+1}{2x \choose x} \cdot \frac{1}{n-x-1}{2n-2x-4 \choose n-x-2}\right)\;.$$ I am told the answer is $$\frac{1}{n}{2n-2 \choose n-1}$$ ...
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I have this algorithm to calculate multiset combinations: $$\mathcal P(k; m_1, m_2, \ldots, m_n) = \Sigma \binom{c(i_1)}{\lambda_1}\ \binom{c(i_2)-\lambda_1}{\lambda_2} \cdots \binom{c(i_s)-\lambda_1-... 1answer 52 views ### what is the probability of bin 1 was chosen Bin 1 contains 20 parts, 5 are defective. Bin 2 contains 15 parts, 4 are defective. One of these Two Bins is chosen at random and 3 parts are randomly selected from the bin chosen. if 2 of the 3 parts ... 2answers 17 views ### Probability of odd-man when n people tossing a coin with probability p of getting head Οdd-man means a person gets a different result from all other people. Probability of getting a head is p. Number of coins knowing the number of people is n. So number of possible ways to get ... 1answer 20 views ### number of subsets of a set with even sum using combinatorics or binomial Let S={a1,a2,a3.......aN}.There are 2^N subsets of this set so if we don't consider the empty set we are left with 2^N-1.We do need to consider cases where it number of odd numbers may be zero and ... 1answer 64 views ### Value of sum of binomials: P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N} [duplicate] P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N} I can calculate the value of this equation manually, but there any direct formula for calculating the value of ... 0answers 35 views ### prime numbers - need a help Helow, There is a question about prime numbers. Supposed that I already answer the first section. I try to answer the second section, but if n \neq 2^{k} (for some k from the natural numbers, ... 2answers 113 views ### Solving {98 \choose 30}+2{97 \choose 30}+3 {96 \choose 30}+…+68{31 \choose 30}={100 \choose q} for q {98 \choose 30}+2{97 \choose 30}+3 {96 \choose 30}+...+68{31 \choose 30}={100 \choose q} Find the value of q? Could someone give me hint as how to solve this question? 1answer 30 views ### Binomial inequality problem {k+n-1 \choose k}\times{k+n+1 \choose k} \leq{k+n \choose k}^2 Can anyone help we with this problem: Let a_n={k+n \choose k}  Prove that a_{k-1}a_{k+1}\leq a_k^2 (\forall k) My first idea was using mathematical induction to proof that for every k element of ... 1answer 21 views ### How find the value of (a_0-a_2+a_4-\ldots)^2+(a_1-a_3+ \ldots)^2 using (1+x)^n=a_0+a_1x+a_2x^2+\ldots+a_nx^n? Q) (1+x)^n=a_0+a_1x+a_2x^2+\ldots+a_nx^n then (a_0-a_2+a_4-\ldots)^2+(a_1-a_3+ \ldots)^2 is equals to 1. 12. 0 (zero)3. 2^{n-1}4. 2^n Answer: (4) well this time i am rocked by this ... 4answers 114 views ### Can anyone give a combinatorial proof of the identity {n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+…+(n-m+1){m \choose m}={n+2 \choose m+2} Can anyone give a combinatorial proof of the identity$${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+\ldots+(n+1-m){m \choose m}={n+2 \choose m+2}$$I am finding difficult as n is varying ... 2answers 113 views ### sum of series \sum \limits_{i=1}^{n}\frac{i(i+1)}{2} [closed] Does there exist an explicit formula for the sum of the series$$\sum \limits_{i=1}^{n}\frac{i(i+1)}{2}$$1answer 155 views ### Sum involving binomial coefficients. Prove that$${^{404}\mathrm C_4}-{^4\mathrm C_1}\cdot{^{303}\mathrm C_4}+{^4\mathrm C_2}\cdot{^{202}\mathrm C_4}-{^4\mathrm C_3}\cdot{^{101}\mathrm C_4} =(101)^4$$I tried writing 101=102-1, but ... 1answer 25 views ### Why does \sum_{k\geq0}\binom{-r}{k}p^r(p-1)^k=p^r(1+p-1)^{-r}? For any positive real number r, it is clear that \binom{-r}{k}(-1)^k\geq0 for all positive integer k. The general binomial theorem then implies$$\sum_{k\geq0}\binom{-r}{k}p^r(p-1)^k=p^r(1+p-1)^...
We consider: $$\dfrac{\Gamma(n)}{\Gamma(k)\Gamma(n-k)}\quad\quad[1]$$ for $n,k\in\mathbb{R}$. Is $[1]$ ever an integer, except for the obvious?