3
votes
1answer
52 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.
0
votes
1answer
22 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 ...
2
votes
5answers
59 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?
0
votes
3answers
81 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+. . ...
2
votes
1answer
40 views

On $\Big[\sqrt n \Big]+ \sum_{j=1}^n \bigg[ \dfrac nj\bigg]$

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)
0
votes
3answers
95 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!} + ...
3
votes
2answers
69 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 ...
0
votes
2answers
82 views

Intriguing Equation

How many ordered tuples of 7 integers ${\{x_{i}\}}_{i=1}^{7}$ are there, such that $$\sum _{i=1}^{7}{x_{i}}-\prod_{i=1}^{7}{x_{i}} =6$$ where $1\le x_i\le 8$. I tried taking ${ \{ x_{ i }\} }_{ ...
0
votes
1answer
21 views

Combination and probability problem. GMAT related.

The Carson family will purchase three used cars. There are two models of cars available, Model A and Model B, each of which is available in four colors: blue, black, red, and green. How many different ...
1
vote
1answer
57 views

23,000 sticks & 500 balls

I am trying to solve two questions that relate to self study. I phrased the questions in a way that makes sense to me, so please ask for clarification if my questions are unclear. Situation: 23,000 ...
2
votes
1answer
61 views

How to show $2\sum^{n/2}_{k=0}$ $(\frac{1}{2}-\frac{k}{n})\binom{n}{k} $ = $\frac{1}{2}$ $\binom{n}{n/2}$

How to show: $$2\sum^{n/2}_{k=0}\left(\frac{1}{2}-\frac{k}{n}\right)\binom{n}{k}=\frac{1}{2}\binom{n}{n/2}$$ n:even please could you help with this equality. on page 17: Rivlin, an intro to ...
0
votes
3answers
71 views

Calculating probability with wordings “no more” and “at least”

75% of children have a systolic blood pressure lower than 136 mm of mercury. What is the probability that a sample of 12 children will include: A) exactly 4 who have a blood pressure greater than ...
0
votes
1answer
25 views

Path in a $(6\times 3)$ rectangle

In the following grid (where each movement is either $1$ step rightward or $1$ step upward) find the number of paths from $P$ to $Q$, if the path between $R$ and $S$ is deleted. (figure below) ...
1
vote
1answer
67 views

How many times does Urmi have to write the digit $1$? [closed]

Urmi has not done her assignment. As a result, her teacher makes him write all the numbers from $1$ to $2007$ on a piece of paper (writing each number only once). How many times does Urmi have to ...
1
vote
2answers
44 views

Find number of triplets

How many triples of positive integers $(a,b,c)$ satisfy $a\le b\le c$ and $abc=1,000,000,000$ I tried prime factorizing R.H.S. and solving equivalently, the equation $\alpha$ + $\beta$ + ...
2
votes
1answer
54 views

Simple expression for this sum?

Is there any simple expression for the sum: $$ S = \sum_{n = 0}^{N-1} \frac{1}{a + e^{2 \pi i n / N}} $$ where $ N $ is a positive integer and $ a $ is some real number. It feels to me like there ...
-1
votes
2answers
90 views

Proof for Binomial theorem

I need to prove this general formula $(1+x)^{n} = \sum_{k=0}^{n} \frac{n!}{k!(n-k)!}x^{k}$ And also prove to prove it on example - equivalence of $(1+x)^{5}$ and its expansion ...
0
votes
2answers
71 views

concrete mathematics: josephus problem

I am going through Concrete Mathematics and am having some difficulty understanding this particular spot in the chapter which introduces the Josephus problem. ...
2
votes
1answer
137 views

How quickly does $(1 + \frac{1}{n})^n$ converge to $e$?

The definition of $e$ boils down to the existence of a limit: $$ \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = 2.71828... $$ And we give this constant a special name, "$e$". This limit ...
2
votes
0answers
178 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
2
votes
1answer
70 views

How to prove $\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2i-1}{i+1} $?

How to prove this closed form involving Catalan numbers? $$\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2 \times (2i-1)}{i+1} $$ I have seen this being used here. Not sure how to derive ...
1
vote
1answer
27 views

Number of selection containing at least one of each kind.

From 3 cocoa nuts, 4 apples, and 2 oranges, how many selections of fruit can be made, taking at least one of each kind ? Ans:315 My thought: For any of our selection that contains at least one of ...
1
vote
2answers
52 views

Why can't you solve this probability problem in this way?

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three ...
1
vote
0answers
68 views

Group forming when the group size is equal

This is from Higher Algebra by Hall and knight. To find the number of ways in which (m + n) things can be divided into two groups containing m and n things respectively. The required number is ...
2
votes
2answers
66 views

Induction: $2^n = \sum_{v=0}^{n} \binom{n}{v}$ [duplicate]

I have to prove the following identity for $n \in \mathbb{N}$: $\displaystyle 2^n = \sum_{v=0}^{n} \binom{n}{v}$ Is there a way to show it through induction? Or is there a easier way? My steps so ...
3
votes
2answers
206 views

Sum of derangements and binomial coefficients

I'm trying to find the closed form for the following formula $$\sum_{i=0}^n {n \choose i} D(i)$$ where $D(i)$ is the number of derangement for $i$ elements. A derangement is a permutation in which ...
-3
votes
3answers
37 views

Number of families based on newspaper-reading habits

$2/5$ of families read "RC" and $3/4$ of families read "CR" newspaper. If $40$ families read none of papers and $100$ families read both. Then number of families in colony are. I am sort of ...
2
votes
1answer
71 views

Sum of series ${n\choose 2a}{a\choose 0}+ {n\choose {2a+2}}{{a+1}\choose 1} + {n\choose {2a+4}}{{a+2}\choose 2} + \ldots$

I wanted to check the rationality of the cosine function for some rational multiples of $\pi$. And I found out that, $\cos(n \cdot\arccos x)$ generates a polynomial in $x$ whose co-efficients have the ...
2
votes
3answers
106 views

Show $\binom{k+1}{r}+\binom{k+1}{r+1} = \binom{k+2}{k+1} $

I have been attempting to show $$\binom{k+1}{r}+\binom{k+1}{r+1} = \binom{k+2}{r+1} $$ and my work is $$\binom{k+1}{r}+\binom{k+1}{r+1} = \frac{(k+1)!}{r!((k+1)-r)!} + ...
0
votes
0answers
126 views

factorial moments of hypergeometric distribution

Factorial moment of positive order : $$\mu_k=\mathbb E[X(X-1)\ldots(X-k+1)]$$ $$=\sum_{m=0}^{n}m(m-1)\ldots(m-k+1)\frac{\binom{a}{m}\binom{b}{n-m}}{\binom{a+b}{n}}$$ ...
0
votes
0answers
56 views

Number of solutions for a linear equation when input has finite set of values

Example: $x_1 + x_2 = 1$ given $x_1, x_2 \in \{0, 1\}$ Only 2 solutions exist, for $x_1 = 1, x_2 = 0$ and $x_1 = 0, x_2 = 1$. Now imagine if you had a number of variables: ...
4
votes
2answers
113 views

$4$ women and $2$ men are being interviewed. Find the probability the women will be interviewed first.

My Calculations: $$\frac{4}{6}\times\frac{3}{5}\times\frac{2}{4}\times\frac{1}{3} = \frac 1 {15}$$ Is that correct?
5
votes
1answer
144 views

Simplify $ \sum_{\{x_{ij}|\forall_i \sum_j x_{ij} = \xi_i, \forall_j \sum_i x_{ij} = \eta_j \}} \prod_{ij} \frac{a_{ij}^{x_{ij}}}{x_{ij}!} $

Let $\{x_{ij}\}$ be a finite set of nonnegative integer variables, with $i = 1..m$ and $j = 1..n$. Let $a_{ij}, \xi_i, \eta_j \geq 0$. Here $\xi_i , \eta_j$ are integers, but $a_{ij}$ can be a real ...
0
votes
1answer
176 views

Number of integral solutions to an equation subject to both upper and lower bounds

Find the number of positive integer solutions to $a+b+c+d+e+f= 20$ subject to $1\leq a,b,c,d,e,f\leq 4$. When there is only the lower bound, i.e $1\leq a,b,c,d,e,f$, we can substitute $g= a-1$, $h= ...
1
vote
1answer
173 views

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 ...
3
votes
3answers
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 $1,2,3,4.$
1
vote
3answers
74 views

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 ...
2
votes
2answers
68 views

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 $d$?
3
votes
2answers
62 views

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 ...
2
votes
3answers
978 views

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 ...
2
votes
1answer
99 views

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 ...
3
votes
2answers
62 views

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 ...
7
votes
1answer
167 views

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 ...
2
votes
3answers
186 views

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 ...
3
votes
3answers
69 views

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!
2
votes
1answer
99 views

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 ...
0
votes
1answer
73 views

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 ...
1
vote
1answer
267 views

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 ...
0
votes
2answers
1k views

How many tries to get at least k successes?

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 $k-1$ ...
3
votes
1answer
90 views

Factorial Identity - True or False?

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 ...