# Tagged Questions

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### 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.
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### 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 ...
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### 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?
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### Number of equivalent rectangular paths between two points

I am trying to determine the number of paths between two points. I am representing the paths as a list of steps "ruru" = right -> up -> right -> up For my purposes, we can assume that there will ...
227 views

### How many functions $f:\{1,2,3,4\}→\{1,2,3,4\}$ satisfy $f(1)=f(4)$?

I just need a hint or a way to think a about this problem: $f(1)$ can be $1, 2, 3, 4$ and $f(4)$ can be $1,2,3,4.$
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### How many functions $f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ satisfy $f(1) + f(2) + \dots + f(10) = 2$?

How many functions $f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ have this property: $$f(1) + f(2) + \dots + f(10) = 2.$$ I understand just $2$ functions can be $1$, the rest have to be $0$, in total ...
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### Why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?

Let $P(X)$ be a polynomial over $\mathbb{Z}$ of degree $d-1$ and $n_0$ be some constant positive integer. Then why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
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### Proof multinomial coefficient is always no less than 1 by induction

The goal is to show $$\binom{n}{k}:=\frac{n!}{k_1!k_2!...k_d!}\geq1$$ where $k=(k_1,k_2,..,k_d)$ multiindex of dimension $d$ with $|k|=n$. It is easy to show the case for $d = 1,2$. $d = 1$ is ...
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### Counting 1:1 and onto functions

I'm faced with the following questions: 1) How many functions are there from a set of size 3 to a set of size 5? How many of them are 1-to-1? 2) How many functions are there from a set of ...
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### License plate consisting of 4 letters and 4 numbers

While doing homework today, the following question popped into my head: Can you easily calculate the amount of unique license plates consisting of 4 letters and 4 numbers in any order? It doesn't ...
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### How to solve the non homogeneous equations

I am looking for the proof of the following I have the following equations $x_1^2+x_2^2+x_3^2+....+x_n^2=1$, $x_1+x_2+x_3+........+x_n=1$ $0 \leq x_i\leq 1$ for-all $i$ I believe that the only ...
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### Closed-form expression for $\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}$?

Wolframalpha tells me that $$\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}=0$$ for $k>2$ However I have not been able to come up with a proof and I simply don't see how to do it. Does anyone ...
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### Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Can this identity be derived from the binomial theorem? $$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$$ I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it ...
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### How to transform $2^{n-2}\frac{(2n-5)(2n-7)…(3)(1)}{(n-1)(n-2)…(3)(2)(1)}$ into $\frac{1}{n-1}\binom{2n-4}{n-2}$?

Just an algebraic step within the well known solution for the number of triangulations of a convex polygon!
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### double digit sums 1-99 * 1-99

How many unique answers are there to all the natural whole numbers 1 - 99 multiplied by all the natural whole numbers 1-99? For instance all the single digits 1-9 multiplied by all the single digits ...
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### Ordered set of integers

$\{x_i\}_{i = 1}^7$ is a set of 7 integers that satisfy $1≤ x_i ≤ 8$. How many such ordered sets of $7$ integers are there, such that x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 - x_1x_2x_3x_4x_5x_6x_7 ...
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### Integer Partitions Formulas [duplicate]

Possible Duplicate: Identity involving partitions of even and odd parts. How would I go about to show the following: Let $pe(n)$ be the number of partitions of size n with an even number of ...
The probability $P'$ of getting at least $k$ successes in $n$ independent tries, given probability of a single success $s$, equals one minus the summed probabilities of getting only $0$ to $k-1$ ...
Let $x$ and $y$ be positive integers. Then, is \begin{align} \frac{x^{xy}}{(xy)!} = \sum_{k_1+...+k_x = xy} \frac{1}{(k_1)!...(k_x)!} \end{align} true, where $k_1$, ..., $k_x$ are all positive ...