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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Proof of Binomial Formula Summation - Induction

Not sure how to go about doing this question. It says that finding the derivative of (1+x)^n is useful!
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Selecting committee with replacement

The UOT Society has an Red Committee (RC) consisting of five members and a Blue Committee (BC) consisting of six members. • Assume there are 6-11 members in the Society, represented by n. Also, ...
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Graphing linear, affine, and convex combinations

For the vectors (2, 1) and (1, 3), how would I graph each of the three combinations? Here are my thoughts (sorry might be totally wrong): linear - plane connecting the two points affine - infinite ...
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help selecting a committee with replacement [on hold]

The Penguin Society has an Wine Committee (WC) consisting of five members and a Beer Committee (BC) consisting of six members. • Assume there are n ≥ 6 members in the Penguin Society. Also, assume ...
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1answer
26 views

Formula to determine total coin combinations problem?

This question was asked in an aptitude test and was meant to be solved within 2-3 minutes.I know how to solve it by Bruteforce method, but its time-consuming.So, is there any strategic way/shortcut to ...
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38 views

Math Combination [on hold]

I have 3 bags, each containing 3 balls. Bag 1$\rightarrow$ Contains a $white$, $blue$ and $green$ ball. Bag 2 $\rightarrow$ Contains a $yellow$, $orange$ and $purple$ ball. Bag 3 $\rightarrow$ ...
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1answer
19 views

How many combinations of 3 or less blocks in 6 holes?

My first instinct is something like $\frac{6!}{3!}$, but this overcounts, and is the result if every cube is unique. However, if the cubes are not unique, then really its just which holes are filled, ...
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1answer
26 views

Partitioning elements into sets

How many ways are there to partition $n$ unique elements into $2$ sets? What about for $k$ sets? I am specifically interested in how to calculate this for varying values of $n$. Moreover, what if ...
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Calculate cardinality of 8-digit strings composed of zeros and ones

How can i calculate cardinality of a set made of 8-digit strings composed of zeros and ones? In general, assuming that digits can repeats. My attempt Let $D$ be the domain composed only by one or ...
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32 views

Determining number of combinations on odd dice

Suppose we had four dice with four sides. Two sides have a $\frac{1}{10}$ chance of being rolled, two sides have a $\frac{4}{10}$ chance of being rolled. The dice score points equal to 1, 3, 4, and 6, ...
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How to write a combination that defines all possible edges in a graph?

Given a graph $G=(V,E)$, I would like to define a set that contains all possible edges in the graph where the edges can't be repeated. In other words, if the graph has three nodes $x,y,z$ then I want ...
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How would you figure out the amount of combinations for a set of n elements, where the amount of elements in each combination doesn't matter?

If you had 4 elements, and you wanted to find all possible combinations of those 4, you take the factorial. But, what if you also wanted to consider combinations of 3, 2 and 1 (where you're still ...
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43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
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1answer
39 views

Expected value of prime lottery ticket

Below is a problem I think that I have solved correctly, but cannot seem to get the correct answer. Any help would be greatly appreciated. You pay $\$13.00$ for a ticket. When you buy a ticket, ...
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1answer
15 views

Combinatorics Number of Possible Assignment Combinations

Say I have Group A and Group B Group A needs 1 student and group B needs 2 students. There are 3 students total (A,B,C). What sort of formula could I use to determine the total number of assignment ...
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1answer
30 views

With $m>n$ , In how many ways $m$ men and $n$ women can seat in row for a photograph so that no two women are adjacent? [duplicate]

Given $m>n$ , In how many ways $ m$ men and $n$ women can seat in row for a photograph so that no two women are adjacent? My effort : There are $m-1$ gaps if $m$ men are seated. Now we have to ...
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Function to define how combinations N items can be organized with a certain condition

This is not a factorial only problem If I have 5 items and I wanted to know how many possible ways they could be arranged, the answer is 5! or 120. However my situation is I need to know how many ...
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57 views

combination of stones on chess board $5\times 5$ [closed]

How many ways can place 6 red, 6 green and 6 blue stones on chessboard $5\times5$, that some row or column is all covered stones of the same color? Thank you for all helping.
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35 views

Linear Permutations of $n$ objects

Suppose there are $n$ distinct objects $O_{1},O_{2},O_{3},\ldots,O_{n-1},O_{n}$. We have to find out the number of ways we can arrange them. But, there is a catch. We have to arrange them such that ...
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2answers
26 views

Possible number of throws in nonagonal dice (ie. two nine-sided dice) [closed]

I have two nine-sided die. Or, in other words, a single pair of nonagonal dice. I was hoping someone could help me with two questions…. How many combinations can be possibly thrown (in a single ...
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1answer
49 views

Prove that $nCr = n(n-1)(n-2)\cdots(n-r+1)/ 1\cdot2\cdot3 \cdots r$ is an integer for all positive integral $n$ and for all integers $r \geq 0$.

Prove that $nCr =\frac{ n(n-1)(n-2)\cdots(n-r+1)}{ 1\cdot2\cdot3 \cdots r}$, is an integer for all positive integral values of $n$ and for all integers $r \geq 0$. Can someone please explain it to ...
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3answers
38 views

Difference between number of positive integer solution and nonnegative integer solution

This question may rather be simple to others but I'm struggling to understand it. So for example, in the book it says, if I want to count the number of compositions for number "$7$" (e.g ...
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2answers
42 views

Counting problem - I seem to be counting double

If we have a group of $10$ men, and $4$ women, and we want to separate these $14$ people to $2$ groups of $7$ such that each group has at least $1$ women, in how many different ways can we achieve ...
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2answers
15 views

Related question permuatation and combination 1

How to identify the question is permuatation or combination? And below is some question: i cannot solve. Show how to solve it. Thank you.
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1answer
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Probability of choosing an item from set and set iteslf

I hvae a question Two cards are drawn from a pack of well shuffled cards. Find the probability that one is a club and other in King. Solving. We split them into two parts 1- If a king is from the ...
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Combinatorics - too much combinations ? where is my mistake?

Problem : A person has 7 friends . How many combinations exists so that he would be able to invite different groups of ...
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1answer
57 views

Help with developing equation with combinatorial numbers

How can I get from $$k^{4} = a*\binom{k}{1}+b*\binom{k}{2}+c*\binom{k}{3}+d*\binom{k}{4}$$ to \begin{equation} k^{4} = \frac{(24a-12b+8c-6d)k+(12b-12c+11d)k^{2}+(4c-6d)k^{3}+dk^{4}}{4!} ...
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1answer
60 views

convolution, trapezoidal distribution pdf

$X$ and $Y$ are independent random variables. $X$ is equal likely to be any value of $\{0,1,2,\ldots,m\}$. $Y$ is equal likely to be any value of $\{0,1,2,\ldots,n\}$. Define $Z=X+Y$. What's the ...
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38 views

counting and permutation problem

i am having hard time figuring this out: a) john,tom,jessie,sam,michael,and amanda want to split among themselves 100 apples how many ways can they split the apples among themselves, if jessie cannot ...
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1answer
28 views

Distinct vs Identical

In a bag containing 20 balls(6 red), (6 green), (8 purple) We draw 5 balls, put them back in the bag, then draw 5 more. In how many ways can this be done if the balls are considered distinct? My ...
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32 views

Number of permutations where an element must precede another

http://www.iarcs.org.in/inoi/2013/zio2013/zio2013-qpaper.pdf The fourth question. Can anyone explain how to solve it? I need to calculate the number of permutations possible while making sure that ...
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a combinations/counting problem

Six volleyball teams, each of which consists of three men and three women, are competing in a tournament. A reporter wants to interview one person from each team, being sure to have three men and ...
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Counting distributions of $52$ cards among $4$ players without distinguishing suits

Imagine a card deck with no suits (no spades, hearts etc.). So there are 4 identical aces, 4 identical 2's and so on. In how many ways can you distribute these cards to 4 players with each player ...
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1answer
39 views

Understanding binary combinatorial problem

It has been quite some time since I've done permutations and combinations, and I'm attempting to remember the proper way to go about solving this issue (not a homework assignment, more of a thought ...
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1answer
40 views

Round table combinatorics?

Sorry if this is a horrible format to read(first time using this site!). Your friends A, B, C and D are going to sit right next to eachother around a round table at your birthdayparty. You do not ...
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Answer of a question

http://www.iarcs.org.in/inoi/2013/zio2013/zio2013-qpaper.pdf The fourth question. Can anyone explain how to solve it? EDIT: Sorry I didn't know about that. I need to calculate the number of ...
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2answers
25 views

About the rank of a small square matrix

An interesting question which hit me just now: Suppose we have a square matrix, for instance, a 3 by 3 one. Each of its entry is a randomly assigned integer from 0 to 9, then whats' the probability ...
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Combinatorics: Prove the number of matches in a Singles Tournament

I was working my way through some problems in Discrete Maths by Rosen, when I came across the following question: There are x players in a singles badminton tournament Show that there are ...
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Number of representatives from states to from a comittee?

Among the three representatives to a conference from each of the fifty states, either none, one, or two of the representatives will be chosen for a large special committee. How many ways can this ...
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1answer
27 views

How to find the value of the following items summed up together?

How to find the value of the following items summed up together? ...
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1answer
128 views

Prove $1! + 2! + 3! + \ldots + n! =y^3$ has only one solution in the set of natural numbers?

I actually know that the above equation is true for $n=1$ and $y=1$ but am unable to prove it for the entire set of natural numbers. Can anyone please help me solve this in a simple way?
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3answers
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A Question on circular permutations.

20 persons are to be seated around a circular table. Out of these 20 , 2 of them are brothers then number of arrangements in which there will be at least three persons between the brothers is.? SO ...
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131 views

$\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$

I can calculate $\sum_{i=0}^k \binom{n}{i}\binom{m}{k-i}$ using $(1+x)^{n+m} = (1+x)^n(1+x)^m$. I would like to calculate the two sums $\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k ...
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1answer
28 views

Optimal solution for maximum value for product of combination

Suppose we have to choose $mm_1$ items out of $m_1$ and $mm_2$ items out of $m_2$ such that $mm_1 + mm_2 = k$ where $k$ is fixed and known. This also constrains us such that $mm_1 < m_1$ and $mm_2 ...
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1answer
29 views

How to determine number of parametric pairs if parameters have different number of values

I have been following a book on an example to determine number of parametric pairs: There are 7 parameters, each with 8 possible values. To determine pairs, they use the combinatorial number (7 ...
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Solving for $r$ in ${12\choose{r}}=924$

I can solve the equation $_{12}C_r=924$ fairly easily by guess and test because there are so few possible $r$ values, but is there a clean way to solve an equation of this format algebraically? I ...
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2answers
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Confusion in combinatorics

Question (1) The number of different ways in which $10$ telegrams can be distributed to 2 message boys is ____? The answer as per the book is $2^{10}$. But, I think answer should be $10^{2}$. If ...
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25 books - permutation question

Imagine 25 books, 5 groups of books (e.g. maths, biology, history, geography, philosophy...) and all groups has 5 seperate colors of books (black, blue, yellow, red, green). So there are 5 blue books ...
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41 views

How many different Tsuro tiles can exist?

The boardgame Tsuro consists of tiles, which each have 8 entry points. Each tile connects each point to exactly one other point. The game manual claims every tile is unique. The game consists of 34 ...
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1answer
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A store sells $4$ kinds of liquor : rum, tequila, whiskey, vodka. How many sets of $7$ different bottles of liquor can one buy?

Arrange the bottles of liquor in the following order: rum, tequila, whiskey, vodka. Then assign to each bottle of tequila its position number increased by $1$, to each bottle of whiskey its position ...