# 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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### Combinatorial formulas and interpretations

I found that $$\sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a (...
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### Anti diagonal elements of table forming pascal traingle

A function in $k$ and $n$ leads to the formation of this table. The elements in this table are rows of pascal triangle if we look at the anti diagonals elements of this table. They have also been ...
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### $\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$

2$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=2\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$. This is an identity in a note for a class in Markov Processes, but I can't ...
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### A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
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### Problem based on sum of binomial coefficients

Let $m$ be the smallest positive integer such that Coefficients of $x^2$ in the expansion $\displaystyle (1+x)^2+(1+x)^3+.....+(1+x)^{49}+(1+mx)^{50}$ is $\displaystyle (3n+1)\binom{51}{3}$ ...
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### Evaluate the combination of $\sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j}$

Can any one help me please to get the approximate result of this combination problem using asymptotic notation: $$\sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j}$$ Thanks
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### T0 show an equation by using binomial theorem

$$\left(1+\frac{a}{n}\right)^{(n-k)} = e^a \left(1-\frac{a(a+k)}{2n}\right)+o\left(\frac{1}{n}\right)$$ as $n\to\infty$. How the binomial theorem show this above equality? Thank you for your help!
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### Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$

I need to proof this following equality : $$\binom{m+n}{m}=\sum_{i=0}^m \left(\binom{m}{i}\binom{n}{i}\right)$$ This is what I did combinatoric proof: Left : subset with $m$ members from $m+n$ ...
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### Coefficient of product of polynomials.

Suppose we have the polynomial $f(x)$ and another polynomial $g(x)$. How can I find the coefficient of say $x^n$ in the product of the polynomials without actually multiplying. I am not that ...
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### Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
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### How to choose $n$ balls from the bags?

Given $4$ bags A, B, C and D. Bag A contains 'a' number of balls. Bag B contains 'b' number of balls. Bag C contains 'c' number of balls. Bag D contains 'd' number of balls. I have another bag E ...
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### Another Hockey Stick Identity

I know this question has been asked before and has been answered here and here. I have a slightly different formulation of the Hockey Stick Identity and would like some help with a combinatorial ...
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### Newton Binomial Problems

Let there be this binomial: $$(\sqrt{2} + \sqrt[3]{3})^{8}$$ How many rational terms are there in it's development? I tought that the number of terms is given by n + 1 = 8 + 1 = 9, but that doesn't ...