# Tagged Questions

47 views

### Simplify factorials into a combinatorial formula

Is there any way to simplify this into a combinatorial formula? $$\frac{t!(n-t)!}{n!}$$
54 views

### How many different positive integer factors does have?

How many different positive integer factors does $(2^7)(3^4)(7^3)(23^5)$ have? Do we have to do any combinations between the powers here?
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### Number of squares on a rectangular board that are neither in the 4th row nor in the 7th column

A rectangular game board is composed of identical squares arranged in a rectangular array of $r$ rows and $r+1$ columns. The $r$ rows are numbered from $1$ through $r$, and the $r+1$ columns are ...
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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 ...
565 views

### 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" ...
143 views

### 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 $k=n/2$ ...
108 views

### 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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86 views

### Simplify a triple sum

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 ...
### Combinatorial proofs of the identity $(a+b)^2 = a^2 +b^2 +2ab$
The question I have is to give a combinatorial proof of the identity $(a+b)^2 = a^2 +b^2 +2ab$. I understand the concept of combinatorial proofs but am having some trouble getting started with this ...