# Tagged Questions

For questions about the study of finite or countable discrete structures, specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

47 views

### in urn A white balls and B black balls. what would be the probability of taking the 5th ball being white

the problem goes like that "in urn $A$ white balls, $B$ black balls. we take out without returning 5 balls. (we assume $A,B\gt4$) what would be the probability that at the 5th ball removal, there was ...
58 views

### Simplify this equation.

Can I simplify or approximate this equation without sigma and combination? \begin{align} \sum_{i = 0}^n (-1)^i {n \choose i} \frac{{d+1}}{d(di + 1)} \end{align}
169 views

### When adding zero really counts …

Note: Although adding zero has usually no effect, there is sometimes a situation where it is the essence of a calculation which drives the development into a surprisingly fruitful direction. Here is ...
66 views

### Parity of $\sum_{i=1}^{n}\lfloor \log_2(i) \rfloor$

Let, $L=\sum_{i=1}^{n}\lfloor \log_2(i) \rfloor$. Problem: Find $n$ for which $L$ is odd. In other words, find a closed form expression (function) $f(n)$of variable $n$ such that $L$ is odd/even if ...
45 views

### Help me understanding what actually i counted with inclusion-exclusion

I tried to solve following task: Count number of $8$-permutations from $2$ letters $A$, $2$ letters $B$, $2$ letters $C$ and $2$ letters $D$ where exactly one pair of same letters are adjacent in ...
29 views

### Transforming generating functions into algorithms that generate combinatorial objects

I've stumbled upon this paper where they write about random sampling of combinatorial objects. For sampling to be proper one has to know some core numbers (probabilities). However, I'm not interested ...
62 views

### Need to prove that there is a continuous sequence which contains 100 cup of coffee , i.e. a man drinks one cup of coffee at the day.

A man can drink at least one cup of coffee at the day. After one year he drinks 500 cup of coffee. Need to prove that there is a continuous sequence which contains 100 cup of coffee, i.e. a man drinks ...
34 views

### When is a recurrence the sum of the powers of the roots of a polynomial?

Newton's formula allows one to calculate the sum $S_n(P)$ of the $n$th powers of the roots of a given monic polynomial $P$ without finding the roots explicitly. (This works even when the roots ...
33 views

### Arrangement of 12 boys and 2 girls in a row.

12 boys and 2 girls in a row are to be seated in such a way that at least 3 boys are present between the 2 girls. My result: Total number of arrangements = 14! P1 = number of ways girls can sit ...
72 views

### Rewriting product to a binomial

I'm currently researching Wigner matrices. I wanted to calculate the moments of its spectral density. The probability density is $$\frac{1}{2\pi} \sqrt{4-x^2} \text{ for } x \in [-2,2]$$ I have ...
47 views

### Method of integration [duplicate]

We have to find the integration of the following function I tried but got stuck can anybody help me how to proceed . Is there anyother method to solve this
56 views

42 views

### Given N blocks, find the number of unique shapes in a NxN block

Constraints: The blocks must be adjacent to each other. i.e. A pair of blocks must have a common edge or vertex. Any shapes that are formed by flipping or rotating or mirroring should be considered to ...
153 views

### Find general solution for the equation $1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r)$

A positive integer $n$ is called a balancing number if $$1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) \tag{1}$$ for some positive integer $r$. Problem: Find the general ...
25 views

### Distribution for playing n scratcher lottery tickets

If I know the prize distribution for a scratcher lottery ticket (i.e. the various prize amounts and the probability associated with each prize) is there a way to form a distribution for playing, say, ...
33 views

56 views

35 views

### Half from any $2n$ but not $2n+2$

Let $n$ be a positive integer. What is the length of the longest possible sequence of $0$'s and $1$'s such that among any $2n$ consecutive numbers, exactly half are $0$'s, but among any $2n+2$ ...
40 views

### Expected Value for Heads for Unknown Weighted Coin Given Head First Flip

This is a combinatorics problem, and I think it involves expected values and conditional probability, but I don't know how to use them: "A bag contains an infinite number of coins whose probabilities ...
30 views

### code to calculate all combinations [on hold]

I have the follwing problem: I have an array u of length d. The sum of every integer of this array u should be k, and integers are from 0 to k. I want now all possible combination that suffice this ...
543 views

### Number of binary search trees on $n$ nodes of height up to $h$

How can I find the number of binary search trees up to a given height $h$, not including BSTs with height greater than $h$ for a given set of unique numbers $\{1, 2, 3, \ldots, n\}$? For example, if ...
18 views

### Finitely many steps to $n$-stone pile.

I have a combinatoric problem still unsolved: $2n$ ($n$ is a positive integer) stones are divided into $3$ piles. In each step, we pick half of a pile which has even number of stones and move those ...
The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...