# Questions tagged [combinatorics]

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

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### How would I formalize this combinatorics question?

We have workouts, durations, and difficulties and we want to know how many unique combinations of these items we can form. Workout Duration Difficulty Strength 10mins Beginner Yoga 15mins ...
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### Probability of rolling any number 10 times at any given point in the sequence [duplicate]

Imagine a race, where each runner has a number, the race track is 100 meters, and each time you roll a fair 6-sided die, the corresponding runner moves 10 meters. I would like to calculate the ...
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### Finding combination of bacteria that represents maximal functions encoded by them

I have the following data where rows represent functions and columns represent bacteria. I have 18 bacteria with a total of 160 functions (all funcitons are present in at least one bacteria). I want ...
203 views

### Amount of total valid strings with multiple constraints

There is a string that has five potential letters (A,B,C,D,E). However, we have some constraints: Each unique ordering of the string is a different passcode. Letters can be used more than once or not ...
26 views

### statistical interpretation to the infinite sum defining cosh(x) = sigma(x^2n/2n!)

the term $x^n$ / n! equals the probability of the sum of n uniform random variables between 0 and 1 to be smaller or equal to x. so $x^(2n)$ / 2n! equals the probability of an even number of random ...
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### Number of permutations when there's a pair of identical items [duplicate]

Suppose there's a bag with four balls; color 0 appears once, color 1 appears once, color 2 appears twice. How many ways are there to draw these balls? Having done this by hand, I believe there's 12 ...
171 views

### How many distinct possibilities for the first $k$ elements of a permutation of $n$ items, when some of the items are identical?

Problem If I have to rearrange $n$ items (which some of them are identical), taking the order into account and only selecting the first $k$ items of every permutation, how many permutations do I have ...
102 views

### Combinatorial/probabilistic meaning/analogy for $x^n / n!$

The expression $x^n / n!$ appears in the infinite sum defining $e^x$ and similar terms in the sums defining $\cos(x)$, $\sin(x)$, etc. I would like to know if there is some combinatorial/probabilistic ...
87 views

### Number of configurations in which you can roll 2 6's in a row

A colleague at work gave me a problem to think about. If you throw $n$ dice, how many configurations are there where you get at least 2 6's in a row? I worked a lot with Pascal's triangle and in the ...
1 vote
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### How do I create random teams of people, each person has multiple parameters, and the parameters are equally distributed among teams

As the title suggests, I am trying to divide a group of people randomly into different teams of equal size. However, each person has parameters that go with them (such as age). If there is only one ...
1 vote
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### Question on permutation and combination

I am having a trouble understanding how to answer the following question and what method to use. The final answer is $415800$ Suppose there are 4 committees A, B, C, and D. 11 candidates are randomly ...
1 vote
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### What statistical methods can I use to determine the popularity of item combinations taken from two groups?

Let's say I have two tables, listing projects with the programming languages and frameworks they use: ...
1 vote
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### Book recommendations for Combinatorial Designs

I am well-acquainted to basic topics of experimental designs such as anova, ancova, crd, rbd, lsd, factorial design, robust design etc (as per the book of Dr. D. C. Montgomery). Now as per my recent ...
1 vote
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### Probability of a person correctly guessing at least one number out of the two number another person chooses

Person A randomly chooses a number from 1 to 5 (inclusive) twice, so A ends up with 2 numbers chosen (can be the same number). Person B also makes a random choice from that list (only 1 number). What'...
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### Is there a formula for finding the number of ways 3 dice throws can add to 6?

Consider 3 dice throws: t1,t2,t3 Let E be the event that t1+t2+t3 add up to 6 What is the probability of event E? I know the denominator is an example of ordered repetition. So with n=6 and k=3 there ...
1 vote
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### How to use Stirling's formula of n! in this probability computations in random walk?

I want to compute $\binom{2n}{n} p^n (1-p)^n = \frac{(2n)!}{n!n!}(p(1-p))^n, n=1,2,3...$ By using an approximation, due to Stirling, which asserts $n! \sim n^{(n +\frac12)}e^{-n}\sqrt{2\pi}$ Where ...
1 vote
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### Theory of 'Runs' and probability

I want to know how did author compute the probability of eleven runs if all arrangements are equally probable. My attempt to answer my own question: This example indicates wide applicability of the ...
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### What is the expected value of this process?

There are $n$ piles each containing $a_i$ stones. In a sequence of moves, Alex chooses two neighbouring piles randomly (containing, say, $A$ and $B$ stones) and combines them to create a single pile ...
27 views

### All possible routes from 8 locations

I know that the combinatorics are simple, but I just can't remember and find a solution. So I have 8 locations (A-H). It does not matter the direction. So Route A -> B is the same as B -> A. How ...
1 vote