0
votes
1answer
26 views

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 ...
35
votes
1answer
553 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 ...
1
vote
3answers
73 views

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.
1
vote
1answer
26 views

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" ...
1
vote
2answers
142 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 ...
2
votes
1answer
72 views

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 ...
2
votes
2answers
46 views

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 $k=n/2$ ...
4
votes
6answers
105 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 ...
0
votes
1answer
20 views

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 ...
0
votes
0answers
25 views

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 ...
4
votes
1answer
84 views

An inverse binomial summation.

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}} ...
1
vote
1answer
37 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.
1
vote
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 ...
3
votes
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.
0
votes
1answer
26 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
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?
0
votes
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+. . ...
-1
votes
1answer
48 views

Password Strength

If passwords of exactly $8$-characters are used, and the character set consists of just lower-case alpha (a-z), how many passwords are possible? Expand the character set to include (A-Z), (a-z), ...
3
votes
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)
0
votes
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!} + ...
3
votes
2answers
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 ...
0
votes
2answers
86 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
29 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
59 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
62 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
89 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
26 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) ...
0
votes
1answer
72 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
58 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
56 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
133 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
96 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
141 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
230 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
76 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
37 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
58 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
73 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
224 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 ...
2
votes
1answer
73 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
107 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
142 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
59 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
137 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
146 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
195 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
236 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 ...
4
votes
3answers
248 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 $1,2,3,4.$
2
votes
3answers
75 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 ...