# Questions tagged [combinatorics]

Counting or enumerating elements in a set or other finite discrete structure.

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### Randomly selecting integers with prescribed minimal distance and estimations

I am interested in sequences of $M$ distinct integers in $[[1,N]]$ (integers from $1$ to $N$) such that integers $I_m$ are separated by at least $\delta$ integers (taking into account the outer ...
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### Drawing the same lottery numbers twince in a row - is it really different? [closed]

While for every single lottery draw the chance is always the same, I know that to get the very same result in the next draw, the probability is squared (1/probablity*probability). But I just cannot ...
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### Optimizing production program

As part of the production of masscustomization products, I have to solve the following problem. A fixed set of sequences (changes from one product to another) is associated with a percentage value of ...
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### Question in combinatorics

I would like to ask a simple question that happened to me on an interview. The question was very simple and it was as follows: We have 80 balls. We have 4 random groups (each group will have 20 balls)...
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### Binomial coefficients - order matters in one subset of events

Question from 'Introduction to Probability' by Blitzstein-Hwang: Two chess players, A and B are to play 7 games. Scores are: a win is 1 points, a draw is 0.5 points and a loss is 0 points. a) How ...
71 views

### the probability of random subset

I'm trying to solve this puzzle but I get stuck. I thought about trying to use the law of total probability to solve intermediate problems with subset of size k but it didn't helped me that much. Is ...
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### Why is the permutations of 6 objects where 2 are similar and the other 4 are similar is the same as 6c2?

I'm studying binomial expansion, and to get all the outcomes or arrangements of head appearing twice after throwing a coin 6 times without using a tree diagram, it's written that we can use 6c2 to get ...
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### Expected Value of Total Number of Three Consecutive Colored Balls

Suppose we have 52 balls in random order where 26 are colored blue and 26 are colored red. The total count of a triplet of red balls in order or whenever a red ball directly follows two previous red ...
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### Grouping classes of objects into more classes?

On page 15 of this Statistics textbook, "Mathematical Statistics and Data Analysis" by John A. Rice, the latter half of Example L has caught my attention: ...
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### Which theory explains this estimation problem?

We have a bag with 100 balls, each labeled 1 to 100. Each ball has a color. 10 are green, 20 are red, 20 are yellow, and 50 are blue. Now someone continuously draws balls from the bag, telling you ...
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### Permutations with exclusions - phone numbers problem

Problem from "Introduction to Probability" by Blitzstein-Hwang: a) How many 7-digit phone numbers are possible assuming phone numbers can't start with 0 or 1. b) Re-solve a) but now phone ...
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### How many ways are there to select exactly one heart in a hand of 5?

I was thinking about it in two ways: choose(13, 1) * choose (39, 4) = (13 * 39 * 38 * 37 * 36) / 4! Slot Method: 13 * 39 * 38 * 37 * 36 then divide out the repeats by dividing by 5! (number of ...
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### Expected value of the largest item in a multinomial distribution

In this question, I use the notation on wikipedia. Suppose $X=(X_1,\ldots, X_k)$ follows a multinomial distribution with parameter $n,\mathbf p$, where $n$ is the sample size or the number of trials, ...
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Copied below is a question that I recently tried to solve from a website: "There are 5 black beads and 5 white beads in your pocket. You pull the beads out of your pocket one at a time, at random ...
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### Understanding combinations and down sampling strategies

Let’s say I have 27 definitions which can define a sample and I have 36 samples which must be defined. The order of these samples does not matter. For instance, one solution may be: 18 objects have ...
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### Equation for no. of unique distributions of k objects, say balls, out of n total, comprised of multiple colors of varying amounts?

Say you have 4 orange balls, 3 red and 2 blue, and you take 6 at a time. I can manually that there are 9 possible distributions: 4o2r, 4o2b, 4o1r1b; 3o3r, 3o2r1b, 3o1r2b; 2o3r1b, 2o2r2b; 1o3r2b. But ...
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### Probability problem confusion

Hello i have a problem and I'm not sure why my solution is wrong. Problem There are 90 students. We're gonna split them in 3 groups, 30 students each. Find the probability that 2 given students A and ...
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### What is the probability that you pick the winning numbers of lottery?

A state runs a lottery called “Pick 6.” A player buys a lottery ticket showing 6 of the numbers from 1 through 40. Then a set of 6 numbers is chosen as the winning numbers, and anyone whose ticket ...
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### How to compute FN and TN explicitly?

Suppose that we have the following clusters ( output of some clustering algorithm ) based on 3 classes : Source : Source link The source explains how to calculate TP+FP and TP explicitly as following ...
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### Lottery Machine Series Matching

Assume there is a lottery machine with 9 slots and 11 symbols (A,B,C,D,E,F,G,H,I,J,K) with decreasing probabilities. In this case, the probabilities are: [0.125,0.125,0.105,0.105,0.1,0.1,0.085,0.085,...
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### sampling without replacement from a population

While counting number of ways to select a sample of size n, from a small(<=60 elements) population, we use combinations to count. For example, number of ways to select a sample of size 3 from a ...
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