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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Formula for particular combinations for array of N binary vectors

I am trying to get to a formula that finds a particular combination Q of N number of binary vectors in an array among all possible combinations of N number of binary vectors, i.e. ...
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2answers
20 views

Number of multisets with restrictions on specific element count

I am looking to find the number of multisets with restrictions on the number of specific elements. This isn't for homework, it is a work related problem. My set of items is {A, a, B, b}. I want to ...
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2answers
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allowable arrangements of ${1,2,2,3,3,3…}$

You are given the multiset $\{1,2,2,3,3,3,\ldots\}$ with each type$(n)$ having $n$ elements.If you pick the first $k$ types, how many allowable arrangements are there using $k$ elements? For $k=5$ ...
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2answers
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How many possible combinations/permutations?

I have 104 ingredients, and there are a maximum of 3 ingredients, how many recipes can be made? Take into account the order matters, that makes this a question of possible permutations instead of ...
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1answer
27 views

How many solutions are there to an n by n queens problem?

Is there a way to calculate the number of solutions to n by n queen problem(done by backtracking) or it's complex and already defined as in the following table
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1answer
33 views

There are $6$ types of cookies. How many different packs of $3$ cookies can the baker package?

A cookie baker packages cookies $3$ to a pack. The type of cookies she can choose from include chocolate chip, oatmeal, sugar-coated, sugar-free, peanut butter, and hazelnut. How many different packs ...
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2answers
23 views

Simple combination / permutation question, any number of items

Apologies if my terminology is incorrect, as I don't have a mathematics background. I'm looking for a formula to work out the number of combinations in a sequence - but where any number of the items ...
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5answers
747 views

How many possible “words” can be made from the first seven letters of the alphabet, allowing for repetition and enforcing alphabetical order?

Using letters from the alphabet $A = \{a, b, c, d, e, f, g\}$, how many words of length $5$ are possible when repetition is allowed but the letters must occur in alphabetical order? Not sure how to ...
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1answer
70 views

Number of sequences that maintain a property

In how many ways can i create a sequence of $m$ elements from the set $1,2,...,n$ such that the longest strictly increasing subsequence of it is exactly $n$? For example if $n=3$ and $m=4$ then the ...
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2answers
24 views

Formation of Teams in Permutation and Combination

A class has $n$ students , we have to form a team of the students including at least two and also excluding at least two students. The number of ways of forming the team is My Approach : To include ...
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1answer
63 views

What is the probability of two-pair poker hand?

To start with, this question has never been asked as how I am going to ask: What is the probability that a five card poker hand will have two pairs (with no additional cards)? Example of two-...
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1answer
42 views

In how many ways can $5$ Indians, $4$ Chinese, and $3$ Americans be assigned to $12$ stations so that no two Americans serve at consecutive stations?

On a railway route from Delhi to Jaipur there are $12$ stations . A booking clerk is to be deputed for each of these stations out of $12$ candidates of whom $5$ are Indians , $4$ are Chinese and the ...
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1answer
36 views

counting number of steps using permutation-combination

We need to climb 10 stairs. At each support, we can walk one stair or you can jump two stairs. In what number alternative ways we'll climb ten stairs? How to solve this problem easily using less ...
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2answers
52 views

How many length-$k$ strictly decreasing sequences where sum is $N$?

How many strictly decreasing sequences of length $k$ in positive integers can I find where the sum of elements is $N$? The problem can be described this way too, I have a number $N$ . Now I want ...
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0answers
17 views

Combinations over tree nodes

Assuming to have a generic tree, how can I calculate all the possible combinations of 1,2,3...n nodes (with n that represents the number of nodes at a certain level of the tree) that can be generated ...
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2answers
31 views

Distinct digits in a combination of 6 digits

How many 6-digit numbers contain exactly 4 different digits? My approach is: For any 3 digis same and the remaining 3 different(aaabcd) 4*9*8*7*6 For any 2 duplicate digits(aabb) and the remaining ...
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2answers
66 views

How many 6-digit numbers contain exactly 4 different digits? [duplicate]

my solution is----> NUMBER can be 777210 this i denote by aaabcd ------ this can be done in ---> 10*1*1*9*8*7*[6!/3!] {1 for a thrice} NUMBER can be 772210 this i ...
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6answers
37 views

Arrangement of 12 boys and 2 girls in a row.

12 boys and 2 girls in a row are to be seated in such a way that at least 3 boys are present between the 2 girls. My result: Total number of arrangements = 14! P1 = number of ways girls can sit ...
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1answer
17 views

What is the equation for the number of combinations of 4 cubes that can be rotated on all axes

I have been trying to work out the number of possible unique combinations of 4 cubes where they can be rotated on any axis. So for example if all the faces of all the cubes where unique across the ...
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1answer
22 views

Concept of alike in Permutation and Combination

Number of ways in which $7$ green bottles and $8$ blue bottles can be arranged in a row if exactly $1$ pair of green bottles is side by side . (Assume all bottles to be alike except for the colour). ...
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1answer
39 views

distributing numbered balls with duplicates into 4 boxes [closed]

How many ways are there to distribute 52 balls, numbered 1 to 13 with 4 duplicates for each number, into 4 distinguishable boxes.
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3answers
31 views

code to calculate all combinations [closed]

I have the follwing problem: I have an array u of length d. The sum of every integer of this array u should be k, and integers are from 0 to k. I want now all possible combination that suffice this ...
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1answer
33 views

How to find number of integral solutions, containing large number of cases?

Number of positive unequal integral solutions of the equation $x+y+z=12$ can be found out knowing the cases it involves: $(1, 2, 9) , (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5)$. Thus, ...
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1answer
45 views

Combination Problem : $6$ Countries , $4$ players from each country

$6$ Countries participate a world tournament . Each country has $4$ players. One Cricket player , One Rugby player , one Volleyball player and one Football player. Need to select a team of $8$ ...
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28 views

Calculating intersection cardinalities of cover sets

I'm having trouble automating calculation of intersection cardinalities of particular sets. Here are some definitions. Number of available elements is $n$, size of a particular set $S \in \...
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1answer
53 views

How many different 4 letter words can be selected from the word ADVANCED?

My attempt : $A-2 , D -2 , V - 1, N -1 , C -1 , E -1 $ $XXXX$ words $=0 $ $XXXY$ words $=0 $ $XXYY$ words $= \binom{2}{2}\times \frac{4!}{2!2!} = 6$ $XXYZ$ words $= \binom{2}{1}\times \binom{5}{...
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1answer
54 views

Is there symbol to denote a combination and permutation?

For example, let's say I wanted to denote any arbitrary, $2$ number combination of the letters, A, B and C. So you can have AB, AC, and BC. Say you wanted a way to represent any given combination, is ...
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3answers
93 views

Coefficient of $x^{r}$

I was willing to find the coefficient of $x^{49}$ in the expression $(x+1)(x+2)...(x+100)$ But is there any kind of generalisation of finding the coefficient of $x^r , 0<r<n$ in the expression $...
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2answers
52 views

find the number of permutations of the letters of the word ANTENNA taken 4 at a time? [closed]

I understand how to get the permutation of the word itself, but what does the "taken 4 at a time" mean?
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1answer
26 views

determine the number of poker hands that are better than 2 Aces, 2 Eights, and a 5?

I am not very familiar with the game of poker so I have no clue where to begin answering this question. it is along the lines of using combinations to solve though.
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1answer
23 views

excecutives from 25 student clubs, one male and one female…

excecutives from 25 student clubs, one male and one female from each are attending a workshop on student violence. how many ways can a commitee be set up of 5 men and 7 women if only one male or ...
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1answer
27 views

In how many ways can 10 different things be distributed to 4 persons if 2 are to receive 2 things and the others are to receive 3 things?

I have no idea how to answer this question, I did a lot of research on trying to figure it out but every answer is so different. I would prefer something along the lines of using combinations and ...
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2answers
39 views

How would you work out these combinations?

If there are 16 different ice-cream flavours, how many combinations are there for a two scoop? If there are still 16 different ice-cream flavours, how many combinations are there for a three scoop? ...
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1answer
21 views

In how many ways can a class of eight students be divided into four groups each having two students?

Suppose there are eight students in a class. We have to divide them into four groups each having two students. The question is, in how many ways can it be done? The answer is: $$\frac{8C2 \cdot 6C2 \...
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Excursion and combination problem

From a group of 12 students 8 are to be chosen for an excursion. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the 8 be chosen. ......
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What's the probability of getting $5$ different numbers but not any $6$ when throwing $5$ dice?

I have $5$ dice, I throw them at once. What is the probability of getting $5$ unique numbers, i.e., $1\ \ \&\ \ 2\ \ \&\ \ 3\ \ \&\ \ 4\ \ \&\ \ 5$ in any order BUT NOT any $6$?...
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2answers
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Number of routes

Suppose there is an ant on the point $(0,0)$ that can move one step right ($(x,y)\mapsto(x+1, y)$), one step up ($(x,y)\mapsto(x, y+1)$) or one step diagnolly ($(x,y)\mapsto(x+1, y+1)$). How many ways ...
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Adding digits to make a number prime or composite

Players A and B alternate writing one digit to make a six-figure number. That means A writes digit $a$, B writes digit $b$, ... to make a number $\overline{abcdef}$. $a,b,c,d,e,f$ are distinct, $a\...
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1answer
29 views

A dance class consists of 22 students, 10 women and 12 men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?

This is a question from Sheldon Ross. A dance class consists of 22 students, 10 women and 12 men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible? So ...
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2answers
29 views

Combinations and permutations with constraints

In how many ways can 6 people be seated around a table for 6, if 2 of the group are always: together? - I solved this by doing 2! x 4! separate? - I do not know how to do this, the answer is 72 ...
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1answer
14 views

Combinations and permutations help

In how many ways can 4 women and 3 men be arranged in a circle? In how many ways can this be done if the tallest woman and shortest man must be next to each other? Ans: 240 I did the first question ...
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5answers
58 views

Combinations Proof [closed]

Show that $2^n = \sum_{i=0}^n \binom ni$. I tried expanding $\sum_{i=0}^n \binom ni$ into sum of: $\frac{n!}{(n-r)!r!}$, and found the common ratio and then used the sum formula, but I did not get to ...
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0answers
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If there are 10 parents and 8 teachers nominated for positions on the school council, how many different committees can there be?

A school council consists of 12 members, 6 of whom are parents, 2 are students, the Principal and the remainder are teachers. The school captain and vice-captain must be on the council. If there are ...
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2answers
73 views

Supposedly really hard problem involving combinations

This problem gives 7 (max) out of 100 points for a college entrance exams. Seems odd because it looks easy to me, although my combinations are not too good. There are $10$ people forming a ...
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Combinatorial analysis

There are $20$ children in a lost ship. They do not remember their birthdays but would like to be assigned with one. 1. In how many ways this can be done so that exactly $2$ children will get ...
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1answer
28 views

Permutation and Combination (high school level) [closed]

In how many ways can 4 Physics books and 3 Mathematics books be arranged on a shelf if a selection is made from 6 different Physics books and 5 different Mathematics books? In how many of these ...
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1answer
19 views

When arranging numbers and letters in combinatorics, should one use multiplication or addition?

Let's say that we are given that a code is formed with 3 letters of alphabet followed by 3 digits from 0-9, and both can be repeated. When required to find the total number of combinations. Is it ...
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22 views

How many combinations to break a monoalphabetic substitution

Let a language $\Sigma$ have 16 letters, we have a message in that language that was encrypted using monoalphabetic substitution (a permutation of the alphabet) and we want to break it. We also ...
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1answer
12 views

Quantifying Onside Kick Variations

I work with a high school football team. We have six (6) different methods/ways we can execute an onside kick (end-over-end, pop-up, jelly roll, copter, squib and drop-kick), five (5) directions we ...
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Number of ways to arrange $n$ numbers based on their relative values to each other

EDIT I've found a formula to solve this question, but I don't understand the reasoning behind it. Can someone explain this formula? $s(n - 1, x + y - 2) \times C(x + y - 2, x - 1)$ $s$ being ...