# Tagged Questions

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### Finite sum equaling Kronecker Delta

could anyone help understand how $$\sum_{j=0}^{n-r}\binom{n-r}{j}*(-1)^{j} = [1 + (-1)]^{n-r}$$ I see that if $j=0$, i get $1=1^{n-r}$, and if $j=n-r$, i get $(-1)^{n-r},$ but what about the rest of ...
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### What is that curve that appears when I use $\ln$ on Pascal's triangle?

I made a little program that generates Pascal triangles as images : I first tried it associating to each pixel a color whose intensity was proportional to the number in the Pascal triangle The ...
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### Product in terms of $n$ of $\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$

What is the following product in terms of $n$? $$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$$ Thank you.
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### Unifying polynomials

Solving a combinatorial problem I find that there are $p(n)=\frac{1}{24}(5n^3+3n^2-2n)$ solutions for even $n$ and $q(n)=\frac1{24}(5n^3+3n^2-5n-3)$ for odd $n$. Now I would like to find a "uninomial" ...
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### Finding a proof to the 'squares' problem

I am trying to find a proof for the general case of the solution to the 'Squares' Problem. This is what I have managed to figure out: If n is the number of squares in the top row, then the number ...
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### Not able to solve this algebra problem.

I tried it, but didn't get anywhere: The real numbers $z_1,\dots ,z_{2011}$ satisfy $z_1 + z_2 = 2z'_1 ,\hspace{1cm} z_2 + z_3 = 2z'_2 ,\hspace{0.5cm} \dots , z_{2011} + z_1 = 2z'_{2011}$ where ...
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### Prove that $\binom {n}{k} = \frac {n!} {(n-k)!k!}$, viewed as a function of $k$, has maximum at $k=\lfloor n/2 \rfloor, \lceil n/2 \rceil$.

Prove that the binomial coefficient $\binom {n}{k} = \frac {n!} {(n-k)!k!}$, viewed as a function of $k$, has maximum at $k=\lfloor n/2 \rfloor, \lceil n/2 \rceil$ if $n$ is odd and maximum at $k=n/2$ ...
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### Finding the coefficient on the $x$ term of ${\prod_{n = 1}^{20}(x-n)}.$

I am trying to find the coefficient on the $x$ term of $\displaystyle{\prod_{n = 1}^{20}(x-n)}$. The issue is that the binomial theorem can't be applied since our $b$ value is changing from term to ...
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### Calculating number of functions

$f$ is a map defined on the set $\mathbf{F}_p$={0,1,2...p-1} to itself. The properties of $f$ are as follows: $f(x)\ne x$ for all non-zero $x$ from $\mathbf{F}_p$. There is exactly only one ...
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### How to measure how many % is done

Let's say that 100 people are going to two exams. They must pass both. In the first exam, 20% of people go forward to the next exam. In the 2nd exam, 50% of people complete and therefore complete the ...
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I am looking for a closed form for this summation: $$\sum_{j=1}^m\frac{r^{-j}}{j{m\choose j}} = \sum_{j=1}^m\frac{r^{-j}}{m{m-1\choose j-1}} = \frac1{rm} \sum_{k=0}^{m-1}\frac{r^{-k}}{{m-1\choose k}} ... 1answer 35 views ### Rearranging asymptotic notation If a \le b^{\frac{1+\log_{2}b}{2}}(1+o(1)), then what is b in terms of a? Whenever I try to rearrange this, I get in a huge mess... Any help would be appreciated. Thanks. 1answer 17 views ### Combination problem approach when at least one of one type must be included Question:In how many ways can 6 be chosen from 4 officers and 8 privates to include at least 1 officer? The correct answer is to consider all the cases in which at least 1 officer is chosen and then ... 1answer 56 views ### How to evaluate this sum 2? \displaystyle\sum_{x+y+z=2014}xy^2z^3 \quad , x,y,z\in\mathbb{N} I think it maybe use combinatorial method. 1answer 25 views ### what is the number of possibilities I have 9 variables that can vary each from 0 to 100.(natural number). And the sum of the first 3 should be between 20 and 30. And the sum of the 9 variables should be equal to 100. What is the number ... 5answers 61 views ### How to find the coefficient of x^8 in \prod\limits_{i=1}^{10}{\left(x-i\right)}? How to find the coefficient of x^8 in (x-1) (x-2) . . .(x-10). Is there any general formula to solve this kind of problems? 3answers 83 views ### 1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1 This is a previous AIME question. 1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1. Then what is a_{17}? Is anything wrong with the following method? 1-x+x^2-x^3+. . ... 1answer 56 views ### On \lfloor\sqrt n \rfloor+ \sum_{j=1}^n \lfloor n/j\rfloor How do we prove that \Big[\sqrt n \Big]+ \sum_{j=1}^n \bigg[ \dfrac nj\bigg] is an even integer for all  n \in \mathbb N ? (where \Big[ \space \Big] denotes the "greatest integer" function) 3answers 97 views ### How to algebraic proof? Need help trying to prove this problem algebraically.$$\binom{n+m}{2} = nm + \binom{n}{2} + \binom{m}{2}$$The farthest i've got is simplifying the RHS to$$nm + \frac{n(n-1)}{2!} + ...
I need to find a closed form for this summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m}\choose{k}}}{j{m\choose j}}r^{k-j+i}$$ I posted this a long time ago, but today I found out ...