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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Bound the probability that the number of red balls in a random partition of a set of balls into 2 equal sized sets is close to the mean.

There are $2N$ unique balls, $2K$ of them are red, the rest are white (thus $2K \le 2N$). You divide the $2N$ balls randomly into two sets, each containing exactly $N$ elements. Is it possible to ...
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115 views

Partition counting problem with cap on pairwise intersection

Fix $T_1,\ldots T_m$ as pair-wise disjoint $k$-subsets of $\{1,2,\ldots,m\cdot k\}$, so that $|T_i|=k$ and $|T_i\cap T_\ell|=0$. For any $j\le k$, how many sets of the form $\{C_1,\ldots,C_m\}$ are ...
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The probability that exactly $k$ balls are in the first urn.

Suppose that $n$ balls are distributed randomly into $m$ urns. Determine the probability that exactly $k$ balls are in the first urn. Progress: I treated the balls as indistinguishable and the urns ...
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1answer
55 views

Jar and Ball Probability Distribution

If I have 8 jars, each jar contains 5 unique ball types. However, I know that I have 20 unique ball types out there. So, I have balls labelled from B1, B2, B3, ...B20 to put into 5 jars. Let's say ...
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1answer
39 views

Expected number of boxes with a blue ball [closed]

Suppose $N$ blue balls, $N$ red balls and $N$ yellow balls are placed in $N$ boxes such that each box has $3$ balls. What is the expected number of boxes with a blue ball? I know it's going to ...
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1answer
34 views

Probability of white ball appearance

There is a basket with balls. There are $M$ white balls and $N-M$ black balls. We take out all balls one after another. Find probability of $k$th ball to be white. Okay, let $A_k$ - be event of $k$th ...
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30 views

Given $d>8$ boxes and $n$ balls. . What is $P(A \cup B \cup C)$?

Given $d>8$ boxes and $n$ balls. Consider event $A$=boxes numbered $1,2,3,4$ receive 0 balls.$B=3,4,5,6$ receive 0 balls, $C=5,6,7,8$ receive zero balls. What is $P(A \cup B \cup C)$? ...
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39 views

Picking K counters out of K buckets containing NK counters, N of each different colour, up to N in each

This is a generalisation of a question that recently came up while solving a TopCoder problem. Suppose we have N blue counters, N red counters, N white counters, and so forth, K colours in total. We ...
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1answer
128 views

Number of ways to place $n$ balls in $k$ bins where just the first $r$ bins have less than $m$ balls

How many ways are there to distribute $n$ balls into $k$ bins where the first $r$ bins have less than $m$ balls (each) and the rest of the bins have more than $m$ (each)? Given this solution for ...
2
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1answer
127 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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1answer
159 views

Example of pairwise independent random process with expected max load $\sqrt{n}$. [closed]

Throw $n$ balls into $n$ bins. Each bin is selected uniformly at random but the process is only pairwise independent. Call the maximum number of balls in any bin the max load. Lemma 2 in these ...
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1answer
86 views

Balls and Bins not randomly

I'm trying to solve a problem similar to this one, I understand the answers of that problem, but my problem is Suppose N bins that are ordered and labeled from 0 to N-1. How many bins in mean do I ...
2
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1answer
127 views

Ways to distribute $400$ non distinct balls to $3$ bins such that in all of the bins there are more than $200$ balls or less than $100$ balls

How many ways there are to distribute $400$ non distinct balls to $3$ bins such that in all of the bins there are more than $200$ balls or less than $100$ balls. My attempt: Notice there's only ...
2
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0answers
106 views

Ways to distribute $25$ distinct balls to $40$ bins with constraints

How many ways there are to distribute $25$ distinct balls to $40$ bins such that: At least one bin will have more than one ball. One and only one bin (not predetermined) will hold exactly ...
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0answers
81 views

Possible combinations of N different balls in M identical bins with the same capacity L

For a distribution center I am interested in the number of possible combinations to put N different boxes (all the same size but different content) into M equal bins (containers) with capacity L, with ...
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1answer
74 views

How many different permutations?

Suppose I've n boxes and m different colored balls of different quantities.How many unique permutations can be obtained ? Example : n=2,m=2, with quantities ( A - 1 ball, B - 2 balls) Thus the ...
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1answer
78 views

Probability that P people will have N distinct birthdays

This question is rather difficult to describe clearly, so I will begin with an example. Suppose I have a 365 people in a room. The odds are very low that all these people have different birthdays. In ...
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Matching Question - a derivative of the birthday problem with two types

I am trying to calculate the expected number of matched trades given b buy orders and s sell orders, each on one of N underlying products. I think this can be cast as a balls-in-bins problem: If I ...
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2answers
106 views

Dividing $n+2$ balls to $n$ bins such that no bin is empty

We have $n+2$ balls and $n$ bins, in how many ways we can divide the balls between the bins such that there's no empty bin? (Bins are different and numbered) My attempt: for the first bin we have ...
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2answers
52 views

Probability that colored balls are separated

Say we throw $b$ blue balls and $r$ red balls uniformly into $n$ boxes. The probability that no box contains a red as well as a blue ball is then, by the inclusion exclusion principle: $$p = ...
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3answers
220 views

n balls are thrown indepently into n boxes. What is the probability there is exactly one box empty?

I am struggling to count the total number of possible outcomes, I assume it is easier to count the outcomes ignoring the order of the balls and the numbering of the boxes?
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2answers
126 views

Probability of $k$ collisions

Say we have $m$ buckets. We select a random bucket and put a ball in it, we repeat this $n$ times. In the end what is the probability of having at least one bucket with exactly $k$ balls? I have ...
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1answer
63 views

Birthday paradox derivation: different approach

I usually use randomization in algorithms so I am familiar with basics of probability but nothing much advanced. I have gone through the derivation for Birthday Paradox (Cormen et al) and decided to ...
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1answer
108 views

Distinguishable balls in two types of indistinguishable boxes

The problem is to count the number of ways to distribute $n$ distinguishable balls in $k$ boxes, where $k-s$ boxes are indistinguishable between each other and the remaining boxes $k - (k-s)$ are ...
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1answer
193 views

How many ways can you put 8 red, 6 green and 7 blue balls in 4 indistinguishable bins?

Assume all balls with the same color are indistinguishable. The order in which balls are put in a bin does not matter. No bins are allowed to have the same distribution of balls! For example, this ...
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75 views

Expansion for r-associated Stirling numbers of the second kind

I am looking for a paper or guidance for expanding the r-associated Stirling numbers of the second kind $S_r(n,k)$. $S_r(n,k)$ is the number of ways to partition a set of n objects into k subsets, ...
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2answers
192 views

Are “Balls in Bins” and “Stars and bars” the same?

I want to know that if Balls in Bins and Stars n Bars problems in combinatorics are similar? Can we reduce one into other? How? How can they be mapped to each other?
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Standard problem of Balls and Bins

A general classification of balls and boxes problems in combinatorics in 12 ways. There are 12 categories since 1. Balls may be distinguishable or indistinguishable; 2. Boxes may be distinguishable or ...
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1answer
118 views

About the distribution of balls in bins

Suppose we have $n$ balls and $n$ bins, and consider the following process: at stage $k$, we throw $\ln{n}$ balls into the bins, independently at random. We stop after $n/\ln{n}$ stages, when all ...
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1answer
159 views

Is this probabilistic balls-and-bins problem well-defined and is my solution correct?

Problem definition: There are $n$ bins, labeled with $1, 2, \ldots, n$. Let $X_i$ be a random variable denoting the number of balls contained in the $i$-th bin. The collection of random variables ...
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1answer
748 views

$n$ balls are thrown randomly into $k$ bins - how many are empty?

A large number of variants of this question were already asked here, including these - one, two, which are close, but none seem to answer my question. Assume that $n$ balls are thrown randomly and ...
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250 views

Probability of $10$ random balls filling $7$ bins. Does indistinguishability matter?

I saw this solution to a problem in my textbook: The problem is we have $10$ indistinguishable balls and $7$ distinguishable bins. What is the probability of having at least one ball in each bin? ...
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flipping two coins when compared to a ball and bin situation?

so as i have understood the sample space for the toss of two coins is TT HH TH HT even if the two coins are absolutely the same we can color each one ...
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141 views

How many outputs possible when throwing $m$ balls into $n$ bins?

For example, for $3$ balls and $3$ bins we have: ...
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173 views

Put B balls in C containers. How many combinations have box(es) with exactly 2 balls?

Assume that we have B balls (all the same) and C numbered containers (distinguishable). We want to calculate how many of the total combinations contain exactly 1 container with 2 balls, exactly 2 ...
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100 views

Probability of drawing balls with different colours

In a jar of balls with different colours, can we find a general probability distribution of having n distinct colours with N number of balls grabbed ? Assuming that any colours have an equal chance to ...
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107 views

n bins, m balls and m>n: Probability of at least r bin containing exactly k balls.When bins are numeret from 1…n and ball is equale.

I want to calculate this probability .In this question $N$ bins, $m$ balls: Probability of any bin containing *exactly* $k$ balls. calculate this but I can not understand this calculation.To be more ...
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105 views

balls in bins — waiting time until $k$ bins are occupied

Consider the classic balls in bins problem: we throw balls one by one into $n$ bins independently and uniformly. Define $\tau(k)$ for $1 \le k \le n$ to be the number of balls we have thrown until $k$ ...
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2answers
132 views

A Question on distribution numbers

This is a question from the book Combinatorics -a problem oriented approach which states: Q.1 Find the no. of distributions of a set of distinct balls into a set of distinct boxes, if no boxes can ...
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1answer
424 views

r identical objects into n distinct boxes

The question says that there are r identical balls to be placed in n boxes s.t. $r\geq n$ .How many ways are there such that each box contains atleast one object?The solution is $n+r-n-1\choose n-1$. ...
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1answer
151 views

r distinct objects into n distinct boxes ( ordering of objects in each box matters)

I have a question and solution as well ,but I don't understand it: If r distinct objects are to be distributed into n distinct boxes each box can hold any number of objects and the ordering of objects ...
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2answers
50 views

Probability that the first $2$ balls are white, given that the sample contains exactly $6$ white balls

An urn contains $30$ white and $15$ black balls. If $10$ balls are drawn without replacement, find the probability that the first $2$ balls are white, given that the sample contains exactly $6$ ...
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2answers
171 views

If I randomly throw 10,000 balls to 4,000 buckets, what is the probability that at least one bucket contains more than 10 balls?

If I randomly throw 10,000 balls to 4,000 buckets, what is the probability that at least one bucket contains more than 10 balls? I don't know how to even approach this. A rough number is also ...
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158 views

Probability of assigning balls into buckets, where each bucket has a certain capacity.

I'll start with a specific example of what I am trying to solve: I have eight balls to be randomly placed into four buckets. Buckets #1-3 have the capacity of 2, 2, 3 respectively, while bucket #4 ...
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Distribution of maximum/minimum proportion in a sampling process

I am facing something that can be explained as a balls & urns problem. Suppose you have $B$ black and $W$ white balls inside an urn. They are randomly chosen, one by one, without replacement, and ...
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110 views

Occupancy distribution bounds for $k$ balls in $m$ bins

Suppose we throw $k$ (distinct) balls into $m$ (distinct) bins, and let $B$ count the number of non-empty bins. I am interested in lower bounds on $B$. More precisely, I wish to bound from above the ...
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1answer
154 views

Probability of drawing a run of a specific color from an urn with two colors of balls

I was sent a puzzle involving an urn with 128 white balls and 288 black. If the balls are drawn without replacement until the urn is exhausted, what is the probability that a sequence of 10 or more ...
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1answer
184 views

Expected number of bins with more than one ball

Suppose that $n$ balls are randomly thrown into $N$ bins. We can compute the expected number of bins that contain at least one ball as $E(X) = N(1 - (1 - 1/N)^n)$. Now, suppose that instead we are ...
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3answers
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Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
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2answers
168 views

How many positive integer solutions are there to the inequality $x_1+x_2+…+x_r\le n$?

The original problem is there are $r$ identical boxes and $n$ identical balls. Every box is nonempty. Then how many ways of putting balls in boxes? It is equivalent to the problem of finding ...