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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1answer
38 views

Rolling 1 die 5 times [on hold]

One die is rolled five times. How many different results are possible? Of those, in how many ways can there be exactly 2 rolls of 4?
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1answer
34 views

How many combinations can be available in a $4\times 4$ grid

There is a $4\times 4$ grid along with $8$ blue tiles and $8$ white tiles. How many combinations are there to fill the grid? Please explain.
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0answers
29 views

Combination of Variables [on hold]

This issue came to me while creating modeling. But since it's mathematical problem, so posting here. I've 200 variables and want to make combination of 6 variables. These set of 6 variables will be ...
1
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0answers
18 views

Distributing 3 white and 10 black marbles to 9 distinct boxes.

Question In how many ways can you distribute 3 white and 10 black marbles (identical) to $9$ distinct boxes? My attempt $3$ white marbles can be distributed in ${3+9-1 \choose 3}={11 \choose 3}$ ways ...
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0answers
11 views

Size of remaining search space for Vehicle Routing Problem given a partial solution

The vehicle routing problem is a NP-hard problem that, in its most basic form, involves scheduling routes for v vehicles that have to make n deliveries in total. So a solution (schedule) has the form ...
5
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3answers
73 views

Suppose a city with Three type of coins ?!

in a city we have tree type 1 dollar, 2 dollar, 3 dollar of coins. we want to pay for a 20 dollar product. how many ways we can pay for a 20 dollar product, if the seller has no money and number of 1 ...
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1answer
265 views

How many different ways are there to organize 2 groups?

In how many different ways can you split 10 people into two groups with the same amount of people? My attempt: Since the order in which you choose someone doesn't matter, I chose to calculate the ...
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1answer
22 views

How to make 4608 combinations with these choices? (Probability, permutations/combinations)

This problem has been giving me a lot of trouble... Freeze King claims to offer 4,608 different ice cream cups. A customer can choose from 3 sizes, 4 flavors; a waffle cone, sugar cone, or cup; ...
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1answer
17 views

Questions regarding seating arrangements [on hold]

Consider there are $9$ people and $3$ tables that sits the following way ($5$ chairs, $3$ chairs and $2$ chairs). How many combinations if order matters to the people being seated and no chair can be ...
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0answers
44 views

When will the game end? [on hold]

Two men are playing a game. They have a card deck consisting of exactly 10 cards, numbered from 1 to 10, and all values are different. On each turn a fight happens. Each of them picks one card from ...
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2answers
72 views

Probability in a Restaurant

In a revolving restaurant, there are four round tables each with three seats. How many different ways can $12$ people sit in this restaurant? This is what I think the answer is: $$\binom{12}{4} ...
1
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1answer
17 views

How do I get number of combination for pairs of football teams?

Suppose we have 8 football teams playing each other in 4 matches. How do I find the number of combinations that is possible? E.g. Teams A,B,C,D,E,F,G,H can be in the following matches: Match 1: A vs ...
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0answers
14 views

Number of ways to connect sets of k vertices in a perfect n-gon [duplicate]

This is a copy of my post at Mathexchange.com, as my question is still not fully answered and I really wanna find a solution to this. Feel free to refer to there for useful comments and partial ...
3
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1answer
37 views

Density of linear combination

Let $r_1, \ldots, r_n$ be a set of positive reals. Define \begin{equation*} S = \{a_1r_1+\cdots+a_nr_n : a_i\in \mathbb{N}\}. \end{equation*} Define $\pi(x)= |\{a\in S:a<x\}|$. Is there an ...
0
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0answers
6 views

The Hopf algebra of symmetric functions and quasi-symmetric functions

The algebras Sym of symmetric functions and Qsym of quasi-symmetric functions are Hopf algebras. As we know, Sym and Qsym are subalgebra of the form power series ring Q[[x_1,x_2,...]]. Are they the ...
0
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1answer
35 views

Dividing $n$ identical things into $r$ groups

I was reading a course on Combinatorics where I came across following: The number of ways in which $n$ identical things can be divided into $r$ groups so that no group contains less than $m$ items ...
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4answers
311 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices (dots), in a given perfect $n$-gon, such that no two lines intersect at the interior of the $n$-gon and no vertice ...
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2answers
29 views

Combination Prob : 3 cards drawn from a pack of 52 cards [closed]

If $3$ cards are drawn from a pack of $52$ cards, then how many ways is it possible that all three cards are of different types? My approach: \begin{equation*} {52 \choose 1} \times {4 \choose ...
0
votes
1answer
25 views

How to calculate the number of combinations of $x$ integers, each with a value between $y$ and $z$?

For example, if I have 4 integers, and each can be between 0 and 36, how many combinations are there? If the numbers have appeared before, but in a new order, then this still counts as a new ...
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0answers
35 views

Ice Cream Sundae [closed]

Really stuck on one problem. An ice cream store offers 15 flavors of ice cream. The speciality is a banana split with 4 different flavored scoops. (a) A two scoop sundae allows for any two choices ...
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0answers
23 views

permutations and combinations with 2 values [closed]

I have 10 different params/columns that can have only Y/N values .How to calculate total combinations which are unique ?
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1answer
30 views

How many different ways can you choose a group of 4 people?

You have a total of 9 people to choose from. Of these 9 people you are supposed to create a group of 4. How many different ways can the new group look? This is my reasoning: To the new group, the ...
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2answers
20 views

Calculate different sequence of scores in a Volleyball match

2 teams play a match of Volleyball During the course of the game, each team gets points, and thus increases its score by 1. The initial score is 0 for both teams. The game ends when: One of the ...
0
votes
1answer
30 views

probability of couples. [closed]

There are 120 people, 10 of them marked BB, 60 of them marked Bb, and 50 of them marked bb. Out of these 120 people, we want to make 60 couples. If picked at random, what percentage of these couples ...
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1answer
18 views

How many different ways are there you can make your choices?

Say you have a playlist with 8 hiphop albums, 6 rock albums and 5 pop albums. You then want to create a new playlist with 3 albums from your other palylist. How many ways are there that you can make ...
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1answer
27 views

How many different towers, with regards to colour, can be built?

You are going to build a tower with coloured blocks. There are ten available blocks, of which three are white, two are red, two are yellow, one is green, one is blue and one is black. The tower you ...
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0answers
13 views

Simplest Way to Detail all N Choose R Combinations on Paper

I'm familiar with the formula to figure out how many non-repeating combinations there are in an n choose r situation, but I was wondering what would be the simplest pattern to go about writing all of ...
6
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0answers
87 views

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$?

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$ so that each column and each row has exactly $n $ $1$'s and $n$ $-1$'s ? I tried for cases $n=1 , 2$ but the solutions were ...
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1answer
28 views

How many pairs of digits can you make using 7 digits? (0-6) [closed]

P.S (0,1) is the same as (1,0) and so on.
2
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0answers
28 views

how to solve questions about bridge hands?

1.how many bridge hands contain 3 aces and exactly one two-card suit? 2.how many bridge hands contain 3 aces and exactly one two-of-a-kind? For the first question,I guess the answer is ...
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2answers
99 views

How many ways can a woman polish her nails if she uses one of two colors on each nail?

A woman is preparing to go to a party and would like to have her nails polished. Suppose she wants to use either the light pink or red nail polish on each nail, how many ways can shepolish her nails? ...
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0answers
97 views

Combinatorial algorithm problem of a symmetric matrix.

Given a matrix A of a strongly $k $ regular graph G(srg($n,k,\lambda ,\mu$);$\lambda ,\mu >0;k>3$). The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$. $ ...
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0answers
15 views

how many bridge hands have the king of spades and at least one diamond?

I think the answer is :C(51,12)-C(38,12).Since one card is fixed in this 13 cards.I just need to choose 12 cards from 52-1=51 cards.And rather than listing all the possibilities of choosing at least 1 ...
2
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1answer
21 views

Using Inclusion-Exclusion Principle to find number of ways to distribute envelopes

I have encountered this problem on a past paper: In how many ways can 675 identical envelopes be divided, in packages of 25, among four student groups so that each group gets at least 150, but no ...
2
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1answer
27 views

Combination or Permutation

Q.1)"Find the no. of ways in which $5$ boys and $3$ girls can be seated in a row so that no two girls are together." Q.2)"In how many ways can $5$ white balls and $3$ black balls are arranged in ...
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1answer
37 views

thought were the same combinatorial

I was under the impression that $${52\choose 5!5!5!5!5!} = {52\choose 5}{47\choose 5}{42\choose 5}{37\choose 5}{32\choose 5} $$ Reason i ask is because i was trying to solve a simple number of ways ...
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2answers
48 views

Combination and Permutations: How many ways can an award be given?

Have this Math question which I'm helping my cousin with but struggling to make sense of the answer. Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of ...
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0answers
40 views

Permutation and Combination Problem-word arrangement

There are three pieces of paper.In the three papers ,a string (non-empty) has to be written such that none of the string on any paper is prefix of some other string.Also alphabet size of characters ...
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0answers
12 views

how many bridge hands contain 4 aces,4 kings,4 queens,and one other card?

I think the answer is :C(4,4)C(4,4)C(4,4)C(40,1). For the choosing of 4 A,kings and queens, there's only one possibility.And for the remaining 1 card, I need to choose it from:52-4-4-4=40 cards,which ...
1
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1answer
36 views

Total no of closed loop paths in 3-by-3 grid

Rules for making a closed loop path: The path must pass through all points. The path have to pass each point only once. The path is formed by joining only consecutive points (defined below). The ...
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2answers
32 views

Need a more compact formula

This is a part of solution of a programming contest problem $$\sum_{i=0}^{k} {x-i \choose 2} $$ given $x-i \ge 2$ is always true. for $k=1$,$(x-1)^2$ $k=2$, $(x-1)^2+((x-2)*(x-3)/2)$ $k=3$, ...
1
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1answer
19 views

How many possible strings exist given base string S and character C?

We're given a base string "abcdefghijklmnopqrstuvwxyz0123" of length 30, and a new character "_" of length 1. How many possible ways can our character be placed (or excluded) in the string without ...
0
votes
1answer
23 views

multiset/combination question

I have a bag full of: 7 green rocks, 12 yellow rocks, and 15 red rocks. How many ways are there to reach in and grab 4 rocks? Is the answer 37C34 (37=7+12+15+4-1) or 6C3 (6=3+4-1)...or something ...
4
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4answers
142 views

Simplify the expression (combination and factorial)

Simplify the following expression: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!}$ My attempt: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n+1)!}{3!(n+1-3)!} * \frac{(n-1)! + ...
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vote
2answers
21 views

How many ways to come up with four teammates?

A Math teacher has to choose 4 people for a competition. If there are 6 boys and 6 girls, and the teacher must select 2 boys and 2 girls, how many ways are there? I came up with ${4 \choose 2} \cdot ...
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1answer
26 views

How many ways to line up if daughters are on sides of mother?

If we have a mother, father, 2 daughters and 3 sons lining up for a family photo, and the mother must be between the daughters, how many ways are there for the family to line up? I came up with ${5 ...
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2answers
35 views

Arrangements of sets of k positions in a n-competitors race

Let $E(n)$ be the set of all possible ending arrangements of a race of $n$ competitors. Obviously, because it's a race, each one of the $n$ competitors wants to win. Hence, the order of the ...
1
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1answer
29 views

function of input size N, combination problem [duplicate]

Can someone please elaborate how from $(N+1)+N+(N-1)+(N-2)$ one can get $= 1/2(N+1)(N+2)$? also how to prove that: $(N-1)+(N-2)+...+3+2+1+0 = \frac{N(N-1)}{2} = {N \choose 2}$ ? Thank you!
0
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0answers
13 views

How many bridge hands have exactly two 5-cards suits and a void so that the remaining suit has a run of 3 cards?

How many bridge hands have exactly two 5-cards suits and a void so that the remaining suit has a run of 3 cards? Should the answer be $C(13,5)C(13,5)C(4,2)C(13,3)$.I first choose $5$ cards for ...
2
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0answers
62 views

Tricky combinatorics problem

I'm trying to solve the following problem: You get $15$ free spins on a slot machine, with a $0.01$ chance of re-triggering a further $15$ spins when a certain symbol falls on the centre reel. You ...