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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 ...
Tony Leung's user avatar
-3 votes
0 answers
21 views

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 ...
Rajeev Patel's user avatar
2 votes
1 answer
27 views

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 ...
Dani007's user avatar
  • 314
0 votes
2 answers
55 views

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? ...
Yusuf Patel's user avatar
0 votes
2 answers
71 views

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 $4$...
IllTime00qw's user avatar
-1 votes
0 answers
31 views

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 ...
RTn's user avatar
  • 297
1 vote
1 answer
68 views

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 ...
Ricci Ten's user avatar
  • 522
0 votes
0 answers
39 views

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 ...
adriavc00's user avatar
0 votes
2 answers
76 views

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 ...
Aman RU's user avatar
0 votes
0 answers
40 views

Permutations of groups with exclusive members

Suppose we have a set $A = \{ a_1, a_2, ... , a_m \}$ and we want to make sets of $3$ elements. We will have $m \choose 3$ different sets. Let's call this new combinations set as $B=\{\{a_1, a_2, a_3\}...
adriavc00's user avatar
1 vote
1 answer
101 views

A Proof about Combination

$\newcommand{\posintset}{\mathbb{Z}^{+}}$ $\newcommand{\domain}[1]{\operatorname{dom}\left(#1\right)}$ $\newcommand{\cardinal}[1]{\abs{#1}}$ $\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}$ I would ...
Ziqi Fan's user avatar
  • 1,840
3 votes
3 answers
388 views

A deceiving simple question of combinations about ways of selecting 5 questions with atleast 1 question from each sections

This question has disturbed several nights of my sleep. There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at ...
ADITYA DAS's user avatar
5 votes
1 answer
51 views

7 students in 4 desks

There are 7 students who need to be seated at 4 desks arranged in a row, each desk having 2 seats. The left seat of each desk must always be occupied. In how many ways can this arrangement be done? My ...
Carlos Lopez's user avatar
2 votes
1 answer
55 views

Probability distribution of a number of $m$ combinations from a set $S$ in a combination of length $l \le |S|$

Given a set of $N$ different numbers. I am interested in the probability $P(m,k,l,N)$ that a number of $m$ combinations (without repetition) of this set, each of length $k$, is not contained in a ...
siserman's user avatar
0 votes
0 answers
91 views

Question about Numbers (combination) Numbers 1 to 9 odd and even numbers

First time posting. I am trying to solve this problem. Consider a collection of cards numbered from 1 to 9. 123456789 All nine cards are arranged in a row so that if the numbers are read from left to ...
Wasabi's user avatar
  • 11
0 votes
0 answers
61 views

Distinguishable objects into distinguishable boxes.

So here is the problem. We have $500$ fans of which $200$ Americans,$150$ Polish, $100$ Italians and $50$ French. They are about to travel to Paris for the start of the Olympic games with 5 different ...
Sergio Guri's user avatar
1 vote
1 answer
90 views

What is the probability of getting 2 chosen ranked cards (any suit). Before the dealer, when the cards are dealt alternately starting from the player?

The player chooses two ranked cards. 52 cards are dealt alternately to the player and dealer, ending up with 26 cards each. What is the probability the player matches the two cards (any suit) before ...
Shaun Andrew's user avatar
0 votes
0 answers
17 views

convex combination in 2D plane.

Demonstrating Equality of Rates at Intersection Points on Parameterized Lines: Consider a scenario in an M1M2 plane, where multiple lines are drawn, each parameterized by alpha and t. Here, alpha ...
Rajlaxmi's user avatar
3 votes
3 answers
530 views

Probability of choosing 4 cards whose sum is 5 from a deck of 40 cards

Let's say we have a deck of cards excluding face cards, so cards from Ace to 10. Which of these is the correct way of computing the probability that the sum of 4 randomly chosen cards is equal to 5? ...
ryangosling's user avatar
0 votes
2 answers
36 views

Error in Thinking about Combinations of 9 Digits

I am making an error in my thinking about this problem, and I'd appreciate a clue as to where I'm going wrong. Let's say we have combinations of nine digits 0-9, of which there are a billion, or $10^...
RD Healthcare's user avatar
0 votes
1 answer
44 views

How many elements, in a set of binary strings, has exactly $k$ blocks of repeating character.

I've been thinking about this problem but I cannot find an analytical formula to get the solutions and generalize it to strings of any length. Let $A$ be a set defined as $\left\{a_1a_2a_3a_4a_5a_6\...
Manuel Parra's user avatar
0 votes
2 answers
50 views

How many different 8 characters passwords with 2 upper-case 2 digits 4 lower-case

A web-banking password is always 8 characters long and it always comprises two upper-case letters from the standard English alphabet, two digits, and four lower-case letters from the standard English ...
Mzq's user avatar
  • 254
0 votes
3 answers
58 views

A mapping is selected at random from all defined on set A . If the selected mapping is injective, the probability that only one element maps on itself

Now my attempt : Total mappings $=n\times n\times n.....=n^n$ Since every element maps to only one other element and none is left untouched in injective: Number of injective mappings $=n(n-1)(n-2)........
Aurelius's user avatar
  • 471
0 votes
0 answers
32 views

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 &...
Dave's user avatar
  • 13
0 votes
1 answer
27 views

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&...
Dave's user avatar
  • 13
2 votes
3 answers
268 views

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 ...
rummy rummyrum's user avatar
0 votes
1 answer
69 views

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 ...
Arthur_D's user avatar
4 votes
3 answers
217 views

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 ...
Ramsunit Raja's user avatar
2 votes
3 answers
167 views

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 ...
Gtexx's user avatar
  • 131
1 vote
1 answer
68 views

Count number of possible combinations of $\sum_{i=1}^{n} a_i \leq 10$

If I have $a_1+a_2 \leq 10$, with $a_1, a_2 \in \{0, \, 1, \, 2, \, \cdots, \, 10 \}$: To count the number of possible combinations for $a_1$ and $a_2$ such that $$a_1+a_2 \leq 10\quad\mbox{and}\quad ...
Liszt Morero's user avatar
-1 votes
2 answers
79 views

Combinations with repetition without bars and stars

I struggle to get the formula for combinations with repetitions. I don't like metaphors and analogies widely used in the other explanations like bins/bars/bookshelves. It seems unnatural to me to ...
mohican93's user avatar
0 votes
0 answers
41 views

How to solve this equation? I've been trying to solve it for hours. [duplicate]

$$^{n-1}C_2+^{n-1}C_3+^nC_4+^{n+1}C_5+^{n+2}C_6+^{20}C_6$$ I Can't figure out how to solve it. I tried writing out the combinations' formula and then solving it like a normal equation, but I ended up ...
iejjxbxvnai's user avatar
0 votes
0 answers
21 views

Count the number of Distributions (no adjacent objects should be identical) [General Approach without any sort of data] [duplicate]

This Question may already be on this platform, I am asking the general approach for this problem if the no. of people were variable and no. of colors were variable. Original Question - Five person A,...
Overloaded's user avatar
8 votes
2 answers
988 views

Number of ways a chess king can move from a1 to h8

Assume a king sits on a1 on an 8x8 chessboard. The king is restricted so he can only move up, right, or diagonally towards top-right. How many paths are there to h8. I know this is a duplicate ...
Jack Hueson's user avatar
-1 votes
1 answer
89 views

A Probablity puzzle on exiting from doors. [closed]

A person enters at $A $(see fig.), and after reaching the point of intersection, he chooses a direction randomly (including the possibility of turning around). If the person reaches an exit then he ...
Leibniz-Z's user avatar
  • 1,009
1 vote
0 answers
65 views

General Approach for: m boys and n girls are in a queue, then find no. of ways in which no. of boys ahead of every girl is greater than no. of girls

I already know that 3 similar questions of this type are already asked on this platform, but I have a little different query and that is for a general case considering this sort of question. Link to ...
Overloaded's user avatar
6 votes
3 answers
1k views

Given 4 red, 3 white and 5 black balls. Picking balls one by one without replacement, find the chance that red balls are exhausted first.

We need to find the probability that all the red balls are exhausted before the white or black balls are exhausted. So, we can still pick white or black balls, but we cannot pick ALL of the white/...
TheMultiRounderGamer's user avatar
4 votes
4 answers
348 views

Probability with Permutations and Combinations

Consider the following problem. Consider a random arrangement of the letters in $GOOSE$. What is the probability the arrangement is $OSGOE$. Now, we know, that the probability is the number of ...
James Chadwick's user avatar
1 vote
1 answer
93 views

$4$ Die Sum equals $20$ - How to generalise it to $n$ die [duplicate]

Calculate the probability that when we roll $4$ fair $6$−sided die, the sum of their upfaces is $20$. My Solution: There exist only the following $5$ possible cases where the sum is $20$ in case of $4$...
Devansh Agarwal's user avatar
-1 votes
1 answer
69 views

If $x+y+z=n$ has $(n-1)(n-2)/2$ positive integer solutions, how many does $2x+3y+6z=1200$ have? [closed]

From combinatorics I know that it's combinations formula $\binom{n-1}2$ for $x+y+z=n$. But how can I deduce it for something with coefficients, like $2x+3y+6z=1200$ For context, how similar problem (...
Michael's user avatar
  • 115
1 vote
1 answer
30 views

Generalizing combinatory probability result

I have a deck of $50$ unique cards numbered from $1$ to $50$. I want to compute the probability to see card n°$1$, $2$ and $3$ appearing after drafting $5$ cards. My reasoning is that if only drafting ...
Blackscholes's user avatar
0 votes
2 answers
99 views

Let $S = \{ 1 , 2 , 3 ,\dots,100\} $, then number of non-empty subsets A of S such that the product of elements in A is even is :

I know my solution's wrong but I am not sure what in it is My solution : To find the number of non-empty subsets $A$ of set $S$ such that the product of elements in $A$ is even, we utilize the ...
Dhroov's user avatar
  • 1
4 votes
1 answer
155 views

Is this expression always greater than $1$?

I was working with probability functions for the binomial and hypergeometric distributions and I came across the following expression; $$ \frac{(a x)! (a (n - x))!}{(a n)!} \frac{n!}{x! (n - x)!} \...
Dotman's user avatar
  • 326
0 votes
1 answer
45 views

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 ...
mvfs314's user avatar
  • 2,084
0 votes
0 answers
38 views

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 ...
Dotman's user avatar
  • 326
3 votes
2 answers
280 views

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 ...
Carlos Lopez's user avatar
-1 votes
2 answers
65 views

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 ...
cc111's user avatar
  • 1
3 votes
1 answer
188 views

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 ...
MR LUN's user avatar
  • 123
1 vote
0 answers
81 views

What is the name of the mathematical function that gives all posible sequences without repetition in a set?

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 ...
LucianoSaldivia's user avatar
2 votes
1 answer
83 views

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 ...
Whisper1231's user avatar

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