# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...