# Questions tagged [combinations]

Combinations are subsets of a given size of a given finite set. All questions for this tag have to directly involve combinations; if instead the question is about binomial coefficients, use that tag.

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### Find number of ways to put 15 medals into boxes of capacity 3, 5 and 8.

There are 15 different medals in a drawer. Suppose the medals in the drawer are put into three boxes that can hold at most 3, 5 and 8 medals respectively. Find the number of ways of distributing the ...
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### Is Probability of drawing 6 sequential numbers lower than drawing 6 non sequential numbers [closed]

I read various questions on this topic and each topic explains that the probability of drawing 1,2,3,4,5,6 when 6 numbers are drawn randomly with numbers between 1 to 45 and no repetitions is same as ...
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### Decomposing complete graphs into edge-disjoint spanning trees

Given a complete graph with $N = 2k$ vertices, is it always possible to decompose it into $k$ edge disjoint spanning trees? If so, then is there a general procedure to find these trees? I would also ...
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### Combinations of Groups of People and the Probability of People being in the Same Group?

Question: Ms. Vena has 25 students in her class. For the next lesson, she would like to put the students into groups of 5. What is the probability that Samantha and Daniel are in the same group? ...
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### matrix combination with unique elements across rows, diagonals, and columns

$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \\ 4 & 3 & 2 & 1 \\ 2 & 1 & 4 & 3 \end{bmatrix}$ Now This is a $4 \times 4$ matrix with $16$ elements $4$...
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### Integral / Summation of combinations [closed]

I was looking at the question Summation of even combinations and as a continuation, I was wondering if a closed form expression exists for $\Sigma_{i=0}^k\binom{n}{i}$ for some $k<n$? Similarly ...
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### finding cardinality of binary sequences of length $10$ that do not contain two consecutive zeros

We have $A=\{(a_1,a_2\ldots a_{10})\in\{0,1\}^{10}|~a_i~\text{and} ~a_{i+1} \text{ are not 0 for any }1\leq i\leq 10\}\}.$ What I see is that the cardinality of $A$ can be found using Fibonacci ...
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### Why is this way of calculating probability of getting a combination wrong?

Suppose we have a set with $m$ elements, we know that the combination of $k$ elements is $m \choose k$. I want to know the probability of getting one specific combination. My question is which way is ...
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### 6 rooms 8 people Arrange Combination Problem with constraints

The 8 residents have to be filled in the 6 rooms such that 1 room has maximum of 1 resident and 2nd room has maximum of 2 residents and the other rooms don't have any limitations. Here's my solution ...
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### Binary combinations with special criteria

Let there be a binary value of "n" bits which consists of only "0"s and "1"s. If we pick exactly "r" "1"s of them (and the rest "n-r" are &...
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### Binary combinations - rank and unrank [closed]

Let's consider a binary value of "n" bits (which consists of only "0"s and "1"s). We want to pick exactly "r" "1"s of them (and the rest "n-r&...
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### A game requires 2 players opposite 2 other players, with 6 people available, how many distinct games can take place?

There are 6 people. A game that requires one pair vs another pair to play is to take place - how many unique games can take place? My Way: 6C2 pairs = 15, and 15C2 distinct matche between each ...
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### A question about dice combinations - cube digit pairs

This is question number 90 on Project Euler. I solved it using a computer approach and my answer was 1217, which is correct. I want to solve it using mathematics, but I am facing a problem because I ...
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### Arranging Letters in a string

Given AACBBBDD. Find the number of ways to arrange these letters in a row such that all B are separated from each other and all D are separated from each other. Note: B and D can come together. But B ...
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### Arranging pairs of friends in a row

Context: $3$ pairs of best friends have made reservations to dine together at a long table.In how many different ways can these $6$ people be seated in a row, such that at least $1$ pair of best ...
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### Number of ways to buy an ice cream

Here is the problem: "In an ice cream shop there are seven flavors of ice cream, including chocolate. A person wants to buy a four-scoop ice cream". Since $n=7,k=4$, then the number of ways ...
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### Understanding the Derivation of a Formula Involving Binomial Coefficients and Factorials

I'm studying a formula that involves binomial coefficients and factorials, and I'm struggling to understand how it was derived. The image below is a screenshot from the paper. They are taking the ...
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### Birthday probability similar to birthday paradox

I am having a party with 20 guests and I want to find the probability to share my birthday with exactly one of the guests (depending on the result, I may buy a second birthday cake, just in case!!). I ...
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### How many ways can we distribute 10 difference looking pencils to 7 students, every student should at least receive one pencil? [closed]

To distribute 10 different cards to 7 students, we can calculate the number of combinations. Since the cards are distinct, the order in which they are distributed does not matter. We are simply ...
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### Inconsistency in answer, by counting in two different ways

A Committee of 5 members is to be constituted for investigating the cases of money laundering from the group of officers belonging to the Economics Offencer Wings, Anti-Terrorist Squad and Indian ...
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I've been reading about Combinations, Variations and Permutations, all with and without repetitions. None of those seems to be what i'm looking for. Given a set of 3 elements $(A, B, C)$, there are 6 ...
### Combination Theory solve for case where $x_i \leq 1$
I am studying computer science and take a discrete math modules, right now trying to learn combinational theory. I know how to solve when cases are $\geq 1$ and x's are all positive but not sure how ...