# Tagged Questions

It's about distribution or arrangement of $m$ distinct or identical balls into $n$ distinct or identical bins with all possible combinations.

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### A Closed Form Lower Bound Approximating $p_{n,m,s} = n![z^n]\left(\sum_{k=0}^s\frac{z^k}{k!}\right)^m$

Here, I found $p_{n,m,s} = n![z^n]\left(\sum_{k=0}^s\frac{z^k}{k!}\right)^m = \sum\limits_{\substack{k_1 + \cdots + k_m=n\\0\leq k_i \leq s}} \frac{n!}{k_1!\cdots k_m!}$ as the number of ways to ...
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### Relation of relative numbers of (restricted) ways to distribute identical / distinct objects into distinct bins

If want to know if the following inequality holds for general values of $s \leq n \ll m$. $$\frac{C_0(n,m,s)}{C_0(n,m)} \leq \frac{p(n,m,s)}{m^n}$$ $C_0(n,m) = \binom{n+m-1}{m-1}$ is the number of ...
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### Find the number of unique paths, given a fixed set, that produce a given output.

I'm looking to find the probability of selecting a given number M given a set of N numbers in a bounded range (or potentially in several differently bounded ranges, but I'm starting simple). I'm ...
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### Baseball related problem (balls and boxes)

Thanks in advance for any help! So I am trying to figure out if the number of hits an inning of baseball is random, or if hits tend to come in bunches. To do this, I'm just using a fairly small ...
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### Intuitively, how many balls are in the bag?

You have a bag that contains all white balls, though you have no idea how many. You reach into the bag and pull out 10 white balls. Then, with the help of a red marker, you mark those 10 white balls ...
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### Expected number of different colored balls in the same bin

Let's say I have $K$ bins and big jar of balls containing $X$ balls. The jar actually contains $W$ white balls and $B$ black balls. Now I'll extract one ball from the jar and throw in one of the $K$ ...
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### Number of ways to place $n$ balls in $m$ bins such that no two balls are placed in the same or adjacent bins?

Assume $m$ distinct bins that are placed on a circle. Therefore, each bin has two neighbors that are its adjacents bins. What is the number of ways that one can place $n$ indistinguishable balls in ...
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### Balls and Bins [Raab 1998 proof]

I cannot work out the proof in one of the steps. The following is copied from the original paper “Balls into Bins” — A Simple and Tight Analysis: The case when $n\log n \ll m \leq n \text{polylog}(n)$...
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### Balls and Bins- Heavy Loaded Case: A Tight Formula

I have $m=10^6$ balls and $n=13\cdot 10^3$ bins. I need to know how I can calculate maximum load for each bin. I'm aware of [1], but it's to loose (e.g. max_load: $e\cdot m/n$. And in practice using ...
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### Selecting point of reference when counting

When arranging people (A, B, C, D, E) in seats (1,2,3,4,5), why isn't it same to count with reference to seats eg. (number of people can sit in seat number 1) * (number of people can sit in seat ...
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### a generic ball and urn problem

Generic ball and urn problem: If two balls are randomly placed in three urns, what are the chances the second urn is occupied? In the generic ball and urn problem, since the balls aren't stated to ...
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### I need a help to understand a paper about Balls and Bins and bins max load

I must declare that I have asked a similar question before but I did not get any answer. I do need a concrete example and formula allowing me to determine maximum number of balls in a bin with ...
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### Bayes theorem. Conditional ball draws

You have a white ball in your hand and there is a ball in a black box that has a probability of 50% of being white. You put the white ball you are holding in the box and shake up the box. a. ...
My question is related to this: http://cs.stackexchange.com/questions/49027/balanced-allocation-hash-table-overflow-probability/49030#49030 In [1,2], it is said that if we throw $n$ balls into \$...