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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Abstract Combinatorics

In a library there is a sequence of $n$ books. There is someone that never wants to take books that are neighborhoods of each other. How many possibilities are there, for him, to take $k\le n$ books? ...
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Finding the optimal configuration of units.

Say I have options of selling items in a=7 or b=1 units. What would be a good equation to help me find the optimal way to supply units for a required quantity? E.g. 21 = (3*a) 9 = (1*a)+(2*b) ...
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2answers
17 views

Why use C(n,r) instead of P(n,r) when considering how many strings can be formed in which a specific letter appears before another specific letter?

I am dealing with a problem in which I must determine how many strings can be formed by ordering the letters ABCDE subject to the conditions given. The condition that I am given is that A appears ...
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3answers
38 views

Combinatorics president and votes

There are 5 candidates for presidency and 11 people that can vote at most one of them (so they can decide not to vote). How many combinations of votes are there if no candidate can recieve more than 5 ...
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1answer
32 views

How many ways can we deal a 13-card hand with at least one suit that does not appear?

In dealing a $13$-hand card that with at least $1$ suit that does not appear, I came up with this: We can choose $3$ of the $4$ suits, as in $3 \choose 4$, and then $13$ cards out of the $39$ cards ...
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1answer
21 views

Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
5
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4answers
63 views

Combinatorial proof of summation of $\sum_{k = 1}^{n-1} {n \choose k}= 2^1 + 2^2 + 2^3 +\ldots+ 2^{n-1}$

I am looking for a combinatorial proof for it. I know how to prove it mathematically. Expanding $(1+x)^n$ and replacing $x$ with $1$ will give me the result but I am not able to explain it ...
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1answer
22 views

Choosing a combination of books, under given restrictions.

Mary has on her bookshelf 5 novels, 5 biographies, and 8 textbooks. Mary decides to take three novels and four non-fiction books with at least one of the non-fiction books a biography. How many ...
2
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2answers
29 views

Divisors of $75600$ of the type of $4n+2$

Find the total no. of divisors of $75600$ which of the type of $4n+2$ where $n\in \mathbb{N}$ and $75600=2^4 \cdot 3^3 \cdot 5^2 \cdot 7^1$ Now I think divisors of type $(4n+2)$ should be of ...
2
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1answer
34 views

In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, …, 500$ such that one number is the average of the other two?

Here's the question which I'm struggling with - In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, ..., 500$ such that one number is the average of the ...
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4answers
36 views

Number of arrangements in which no two persons sit side by side

There are $10$ seats in the first row of the theater out of which $4$ are to be occupied. Find the number of ways of arranging $4$ people so that no two people sit side by side. Making different ...
2
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1answer
38 views

solve $54 x + 16 y = 2400$ for integer values of x,y

How to get integer values for x and y that satisfy: $$54 x + 16 y = 2400$$ Someone told me that I can do it using Euclid-Wallis algorithm, but I don't understand it so, if there isn't any else ...
3
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1answer
31 views

A formula for length of representation of a number in a “base” without zeros

If you had 2 items the sequence would go like this: $$1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5, \ldots$$ This is $\lfloor\log_2(n+2)\rfloor$. What if I ...
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1answer
20 views

Calculating number of combinations of multiple sets, each containing different number of elements

I'm not a math genius so please consider that when posting your explanation. I have the following sets, arbitrarily named: a [a1, a2, a3] b [b1, b2] c [c1, c2, c3, c4] d [d1, d2, d3, d4, d5] e ...
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7 views

Circular arrangements with multiples

I was reading the stack exchange circular arrangements problems and was wondering how one would discover the distinct arrangements with multiplies. If there were $9$ individuals which consist of $3$ ...
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1answer
38 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
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1answer
15 views

find all combination without overlapping

(*Constraint) Kinds of number should be limited in 3 (ex. {1,1,1,1} o/ {1,2,3} o / {1,1,2,2,5,5,4} x) And I want to find series of Integer. For example, if n = 4 (n is length of numbers) {2, 7, 2, ...
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2answers
31 views

Find the number of seating arrangements at a round table of three single men, two single women, and two families

Three single men, two single women and two families take their places at a round table. Each of the two families consists of two parent and one child. Find the number of possible seating arrangements. ...
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1answer
40 views

The number of ways to divide 10 people into groups of given size [closed]

Find the number of ways in which $10$ people can be divided into $2$ groups consisting of $7$ and $3$ people Three groups consisting of $4$, $3$ and $2$ people with $1$ person rejected. $5$ groups ...
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2answers
42 views

In how many ways can a committee of $6$ people be selected from $7$ men and $6$ women if it can contain at most one of persons A and B?

A committee of $6$ people will be formed with $7$ men and $6$ women. The oldest of the $7$ men is A and the oldest of the $6$ women is B. It is described that the committee can include at most one of ...
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1answer
34 views

Properties of lists with arbitrary lengths and alphabet size

I am having trouble understanding a problem of applying the concepts of permutations and combinations in an example that I found while reading my textbook. Basically, it wants the number of elements ...
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2answers
47 views

What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is …

What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is $2^33^3$ where $x, y,z\in \Bbb N$? What I tried : At least one of $x, y$ and $z$ should have factor ...
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5answers
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Give a proof of ${n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + … + {n \choose n}^2 = {2n \choose n}$ [duplicate]

I must prove this: ${n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + ... + {n \choose n}^2 = {2n \choose n}$ But, I have no idea how to prove it or how it necessarily works. Could someone help ...
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2answers
54 views

Give a Combinatorial proof to show $\sum_{i=1}^{n}{iC(n,i)}=n2^{n-1}$

I am completely lost on how to achieve this. I have no idea where to start, nor do I know what to use to find to prove this problem. Can someone help me with this?
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2answers
50 views

How many numbers between $0$ and $1,000,000$ have exactly one digit equal to $9$ and the sum of digits equal $13$?

How many numbers between $0$ and $1,000,000$ have exactly one digit equal to $9$ and the sum of digits equal $13$? I am not sure how to start this problem to get this answer. Could someone help me ...
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2answers
36 views

Counting problem (students assigned to a tutor)

Four new students have to be assigned to a tutor. There are seven possible tutors, and none of them will accept more than one new student. In how many ways can the assignment be carried out? The ...
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1answer
32 views

Number of Different Elements of $S_{ijkl}$ with Some Symmetries

I am not good at combinatorics so I am asking this simple question to learn a little. In fact, this question is motivated by the symmetries happening for the stiffness and Eshelby tensors in the ...
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1answer
23 views

How find the length of an array

Story: In fact this question is related to THIS. How to create an array maintaining following conditions- ...
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13 views

To prove a combination problem

Given a series of integers, ${p_1},{p_2},...,{p_m}$, and ${p_i}\in[0, P]$, satisfy the following inequality: $\sum_{i=1}^{m}{p_i}\leqslant P$, How can I prove that the total number of the solutions to ...
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1answer
27 views

26 flavours of ice-cream, how many different banana splits can be made that have 3 different flavours?

A boutique ice cream bar stocks 26 flavours and offers a rainbow banana split that contains 3 scoops of ice cream, each of a different flavour. How many different rainbow splits can the store ...
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0answers
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How many people at the party?

At a party, there are $n$ people. A waiter counts 188 cin-cin. How many people partecipate at the toast? I have solved the problem in this way: $\displaystyle\frac{n(n-1)}{2}=188$ but I ...
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2answers
26 views

Counting ways to arrange the word REGULATIONS.

Find the number of ways the word REGULATIONS can be arranged such that there are exactly $4$ letters between $R$ and $E$ . I did $4!\ \ \ \ \text{for}\ \ ...
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2answers
26 views

Permutation and Combination Committee Questions [closed]

Q1. In how many ways can we select a committee of 6 persons from 6 boys and 3 girls, if at least two boys and two girls must be there in the committee? Given Answer: 65 Q2. In how many ways 7 persons ...
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1answer
25 views

Binomial distribution, explanation formula

I have a really simple question. I can't figure out the meaning of the binomial coefficient in the case of a binomial distribution formula. I know what the formula means, and how to use it for the ...
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0answers
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How can I divide 2 identical objects of one type, 2 identical objects of second kind and 2 identical objects of third kind?

How can I divide $2$ identical objects of one type, $2$ identical objects of second kind and $2$ identical objects of third kind into $3$ groups such that each groups contains only two objects. ...
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1answer
17 views

Discrete Combination Problem

I cannot seem to figure this out: A 8-Person committee is to be founded by a group of fifteen women and twelve men, how many ways can the committee be chosen if: (Already solved->) a)The committee ...
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1answer
83 views

Number of binary numbers given constraints on consecutive elements

I've been trying to solve this question for quite a while, given to us by our discrete maths professor. I've been having a hard time in general with it, so I thought I tried looking it up online but ...
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1answer
99 views

How to partition $nk$ objects $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size $k$, so that no combination of $k$ is repeated.

What is an algorithm to partition $nk$ objects a total of $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size exactly $k$, so that no subset of size $k$ is ever repeated? For example, ...
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1answer
20 views

Generate Combinations of Clusters

My maths background is not so strong, please go slowly. Suppose I have 3 elements, which are A, B and C. I want to find (1) the number of operations to perform to find (2) all possible clusters that ...
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3answers
56 views

$20$ people sit at a round table, how many ways can we choose $3$ with no $2$ being neighbors?

My thought process to this problem was as follows: $1st$ move you have $20$ choices, when you pick you eliminate $3$ people, the first person and their two neighbors. The $2nd$ move you have $20-3$ ...
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2answers
60 views

Ways to create a quadrilateral by joining vertices of regular polygon with no common side to polygon

How many ways are there to create a quadrilateral by joining vertices of a $n$- sided regular polygon with no common side to that polygon? It's quite easy to solve for triangles for the same ...
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2answers
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In how many ways can a dating service match each of six females one of eight males?

The question reads: A dating service has the names of six females and eight males who seek matches. In how many ways can the service match each female with one male? Using the multiplication ...
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2answers
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Combination - Distribution of gifts

Seven different type of gifts are to be distributed among 10 children.Every kind of gift must be at least given to one child. Then, how many combinations do we have? Note:You have A, A, A.... ...
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1answer
22 views

What is the probability of getting two spades in five draws?

For easier viewing, here is the question: What is the probability of getting two spades from five draws I understand how to solve this question using permutations. I have explained my work ...
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4answers
44 views

Why do we use nCk when determining numbers of favorable outcomes of coin tosses?

After delving back into probability a bit, I'm absolutely stumped as to why we would use nCk to answer the question "What is the probability of getting 3 heads when tossing a fair coin 10 times?" I ...
6
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2answers
93 views

Number of solutions $(x_1)(x_2)(x_3)(x_4) = 2016$

Having some trouble wrapping my head around this one: find the number of solutions to the equation $(x_1)(x_2)(x_3)(x_4) = 2016$, where $(x_i)$s are integers that are not necessarily positive. ...
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4answers
41 views

Number of ways to roll five 6-sided dice with sum 7

I would like to determine the number of possible outcomes that are possible to roll five fair $6$-sided dice where the sum of the faces adds up to $7$. I am interested in the case where order does ...
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1answer
33 views

Permutations + Combinations Proof

$W_n^{(k)}$ is the number of permutations in a set of all $n!$ permutations in a $n$-element set which has $k$ fixed points. $W_n^{(0)}$ is the number of n-derangements where $\frac{W_n}{n!} ...
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2answers
30 views

How many combinations of three numbers using 1, 2, 3, and 4 exist?

If you count (4 4 3) as one combination, you cannot count (4 3 4) as another. My approach is $\dfrac{4^3}{3!}$, but obviously this does not work. I don't know why it doesn't work, and I don't know ...
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1answer
54 views

Number of ways to distribute the awards?

Q: There are 25 participant in a contest in which first, second, and third place prizes are awarded as well as 3 honorable mentions. How many ways are there to hand out the top three prizes? After ...